FORMALIZED MATHEMATICS Vol.1,No.5,November–December 1990 Université Catholique de Louvain

Properties of Fields

Józef Białas^{1} University of Łódź

Summary. The second part of considerations concerning groups and fields. It includes a definition and properties of commutative field F as a structure defined by: the set, a support of F, containing two different elements, by two binary operations +_{F}, ·_{F} on this set, called addition and multiplication, and by two elements from the support of F, 0_{F} being neutral for addition and 1_{F} being neutral for multiplication. This structure is named a field if ⟨the support of F, +_{F}, 0_{F} ⟩ and ⟨the support of F, ·_{F}, 1_{F} ⟩ are commutative groups and multiplication has the property of left-hand and right-hand distributivity with respect to addition. It is demonstrated that the field F satisfies the definition of a field in the axiomatic approach.

MML Identifier: REALSET2.

The articles [4], [2], [3], and [1] provide the notation and terminology for this paper. A field structure is said to be a field if: (Def.1) there exists an at least 2-elements set A and there exists a binary op-

eration o_{1} of A and there exists an element n_{1} of A and there exists a binary operation o_{2} of A preserving A \ {n_{1}} and there exists an element n_{2} of A \ single(n_{1}) such that it = field(A, o_{1}, o_{2}, n_{1}, n_{2}) and group(A, o_{1}, n_{1}) is a group and for every non-empty set B and for every binary operation P of B and for every element e of B such that B = A \ single(n_{1}) and e = n_{2} and P = o_{2} _{n_{1}} A holds group(B, P, e) is a group and for all elements x, y, z of A holds o_{2}(⟨⟨x, o_{1}(⟨⟨y, z⟩⟩)⟩⟩) = o_{1}(⟨⟨o_{2}(⟨⟨x, y⟩⟩), o_{2}(⟨⟨x, z⟩⟩)⟩⟩) and o_{2}(⟨⟨o_{1}(⟨⟨x, y⟩⟩), z⟩⟩) = o_{1}(⟨⟨o_{2}(⟨⟨x, z⟩⟩), o_{2}(⟨⟨y, z⟩⟩)⟩⟩).

Next we state the proposition

(1) Let F be a field structure. Then F is a field if and only if there exists an at least 2-elements set A and there exists a binary operation o_{1} of A and there exists an element n_{1} of A and there exists a binary operation o_{2} of A preserving A \ {n_{1}} and there exists an element n_{2} of A \ single(n_{1}) such that F = field(A, o_{1}, o_{2}, n_{1}, n_{2}) and group(A, o_{1}, n_{1}) is a group and for every non-empty set B and for every binary operation P of B and for every element e of B such that B = A \ single(n_{1}) and e = n_{2} and P = o_{2} _{n_{1}} A holds group(B, P, e) is a group and for all elements x, y, z of A holds o_{2}(⟨⟨x, o_{1}(⟨⟨y, z⟩⟩)⟩⟩) = o_{1}(⟨⟨o_{2}(⟨⟨x, y⟩⟩), o_{2}(⟨⟨x, z⟩⟩)⟩⟩) and o_{2}(⟨⟨o_{1}(⟨⟨x, y⟩⟩), z⟩⟩) = o_{1}(⟨⟨o_{2}(⟨⟨x, z⟩⟩), o_{2}(⟨⟨y, z⟩⟩)⟩⟩).

1Supported by RPBP.III-24.

Let F be a field. The support of F yielding an at least 2-elements set is defined by:

(Def.2) there exists a binary operation o_{1} of the support of F and there exists an element n_{1} of the support of F and there exists a binary operation o_{2} of the support of F preserving the support of F \ {n_{1}} and there exists an element n_{2} of (the support of F) \ single(n_{1}) such that F = field(the support of F, o_{1}, o_{2}, n_{1}, n_{2}).

The following proposition is true

(2) For every field F and for every at least 2-elements set A holds A = the support of F if and only if there exists a binary operation o_{1} of A and there exists an element n_{1} of A and there exists a binary operation o_{2} of A preserving A \ {n_{1}} and there exists an element n_{2} of A \ single(n_{1}) such that F = field(A, o_{1}, o_{2}, n_{1}, n_{2}).

Let F be a field. The functor +_{F} yielding a binary operation of the support of F is defined as follows:

(Def.3) there exists an element n_{1} of the support of F and there exists a binary operation o_{2} of the support of F preserving the support of F \ {n_{1}} and there exists an element n_{2} of the support of F \ single(n_{1}) such that F = field(the support of F, +_{F} , o_{2}, n_{1}, n_{2}).

Next we state the proposition

(3) For every field F and for every binary operation o_{1} of the support of F holds o_{1} = +_{F} if and only if there exists an element n_{1} of the support of F and there exists a binary operation o_{2} of the support of F preserving the support of F \ {n_{1}} and there exists an element n_{2} of the support of F \ single(n_{1}) such that F = field(the support of F, o_{1}, o_{2}, n_{1}, n_{2}).

Let F be a field. The functor 0_{F} yielding an element of the support of F is defined by:

(Def.4) there exists a binary operation o_{2} of the support of F preserving the support of F \ {0_{F} } and there exists an element n_{2} of the support of F \ single(0_{F}) such that F = field(the support of F, +_{F} , o_{2}, 0_{F} , n_{2}).

Next we state the proposition

(4) For every field F and for every element n_{1} of the support of F holds n_{1} = 0_{F} if and only if there exists a binary operation o_{2} of the support of F preserving the support of F \ {n_{1}} and there exists an element n_{2} of the support of F \ single(n_{1}) such that F = field(the support of F, +_{F} , o_{2}, n_{1}, n_{2}).

Let F be a field. The functor ·_{F} yields a binary operation of the support of F preserving the support of F \ {0_{F} }and is defined as follows: (Def.5) there exists an element n_{2} of the support of F \ single(0_{F}) such that

F = field(the support of F, +_{F} , ·_{F} , 0_{F} , n_{2}).

We now state the proposition

(5) For every field F and for every binary operation o_{2} of the support of F preserving the support of F \ {0_{F} } holds o_{2} = ·_{F} if and only if there exists an element n_{2} of the support of F \ single(0_{F}) such that F = field(the support of F, +_{F} , o_{2}, 0_{F} , n_{2}).

Let F be a field. The functor 1_{F} yielding an element of the support of F \ single(0_{F}) is defined as follows: (Def.6) F = field(the support of F, +_{F} , ·_{F} , 0_{F} , 1_{F}).

The following propositions are true:

(6) For every field F and for every element n_{2} of the support of F \single(0_{F}) holds n_{2} = 1_{F} if and only if F = field(the support of F, +_{F} , ·_{F} , 0_{F} , n_{2}).

(7) For every field F holds F = field(the support of F, +_{F} , ·_{F} , 0_{F} , 1_{F}).

(8) For every field F holds group(the support of F, +_{F} , 0_{F}) is a group.

(9) For every field F and for every non-empty set B and for every binary operation P of B and for every element e of B such that B = the support of F \ single(0_{F}) and e = 1_{F} and P = ·_{F} _{0_{F}} the support of F holds group(B, P, e) is a group.

(10) Let F be a field. Let x, y, z be elements of the support of F. Then (i) ·_{F} (⟨⟨x, +_{F} (⟨⟨y, z⟩⟩)⟩⟩) = +_{F} (⟨⟨·_{F} (⟨⟨x, y⟩⟩), ·_{F} (⟨⟨x, z⟩⟩)⟩⟩), (ii) ·_{F} (⟨⟨+_{F} (⟨⟨x, y⟩⟩), z⟩⟩) = +_{F} (⟨⟨·_{F} (⟨⟨x, z⟩⟩), ·_{F} (⟨⟨y, z⟩⟩)⟩⟩).

(11) For every field F and for all elements a, b, c of the support of F holds +_{F} (⟨⟨+_{F} (⟨⟨a, b⟩⟩), c⟩⟩) = +_{F} (⟨⟨a, +_{F} (⟨⟨b, c⟩⟩)⟩⟩).

(12) For every field F and for all elements a, b of the support of F holds +_{F} (⟨⟨a, b⟩⟩) = +_{F} (⟨⟨b, a⟩⟩).

(13) For every field F and for every element a of the support of F holds +_{F} (⟨⟨a, 0_{F} ⟩⟩) = a and +_{F} (⟨⟨0_{F} , a⟩⟩) = a.

(14) For every field F and for every element a of the support of F there exists an element b of the support of F such that +_{F} (⟨⟨a, b⟩⟩) = 0_{F} and +_{F} (⟨⟨b, a⟩⟩) = 0_{F}.

Let F be an at least 2-elements set. A set is said to be a one-element subset of F if: (Def.7) there exists an element x of F such that it = single(x).

We now state the proposition

(15) For every at least 2-elements set F and for every one-element subset A of F holds F \ A is a non-empty set.

Let F be an at least 2-elements set, and let A be a one-element subset of F. Then F \ A is a non-empty set.

The following proposition is true

(16) For every at least 2-elements set F and for every element x of F holds

single(x) is a one-element subset of F. Let F be an at least 2-elements set, and let x be an element of F. Then

single(x) is a one-element subset of F.

The following propositions are true: (20)^{2} For every field F and for all elements a, b, c of the support of F \

single(0_{F}) holds ·_{F} (⟨⟨·_{F} (⟨⟨a, b⟩⟩), c⟩⟩) = ·_{F} (⟨⟨a, ·_{F} (⟨⟨b, c⟩⟩)⟩⟩).

(21) For every field F and for all elements a, b of the support of F \single(0_{F}) holds ·_{F} (⟨⟨a, b⟩⟩) = ·_{F} (⟨⟨b, a⟩⟩).

(22) For every field F and for every element a of the support of F \single(0_{F}) holds ·_{F} (⟨⟨a, 1_{F} ⟩⟩) = a and ·_{F} (⟨⟨1_{F} , a⟩⟩) = a.

(23) For every field F and for every element a of the support of F \ single(0_{F}) there exists an element b of the support of F \single(0_{F}) such that ·_{F} (⟨⟨a, b⟩⟩) = 1_{F} and ·_{F} (⟨⟨b, a⟩⟩) = 1_{F}.

Let F be a field. The functor −_{F} yielding a function from the support of F into the support of F is defined by:

(Def.8) for every element x of the support of F holds +_{F} (⟨⟨x, −_{F} (x)⟩⟩) = 0_{F}.

One can prove the following propositions:

(24) For every field F and for every element x of the support of F holds +_{F} (⟨⟨x, −_{F} (x)⟩⟩) = 0_{F}.

(25) For every field F and for every function S from the support of F into the support of F holds S = −_{F} if and only if for every element x of the support of F holds +_{F} (⟨⟨x, S(x)⟩⟩) = 0_{F}.

(26) For every field F and for every element x of the support of F and for every element y of the support of F such that +_{F} (⟨⟨x, y⟩⟩) = 0_{F} holds y = −_{F} (x).

(27) For every field F and for every element x of the support of F holds x = −_{F} (−_{F} (x)).

(28) For every field F and for all elements a, b of the support of F holds +_{F} (⟨⟨a, b⟩⟩) is an element of the support of F and ·_{F} (⟨⟨a, b⟩⟩) is an element of the support of F and −_{F} (a) is an element of the support of F.

(29) For every field F and for all elements a, b, c of the support of F holds ·_{F} (⟨⟨a, +_{F} (⟨⟨b, −_{F} (c)⟩⟩)⟩⟩) = +_{F} (⟨⟨·_{F} (⟨⟨a, b⟩⟩), −_{F} (·_{F} (⟨⟨a, c⟩⟩))⟩⟩).

(30) For every field F and for all elements a, b, c of the support of F holds ·_{F} (⟨⟨+_{F} (⟨⟨a, −_{F} (b)⟩⟩), c⟩⟩) = +_{F} (⟨⟨·_{F} (⟨⟨a, c⟩⟩), −_{F} (·_{F} (⟨⟨b, c⟩⟩))⟩⟩).

2The propositions (17)–(19) became obvious.

(31) For every field F and for every element a of the support of F holds ·_{F} (⟨⟨a, 0_{F} ⟩⟩) = 0_{F}.

(32) For every field F and for every element a of the support of F holds ·_{F} (⟨⟨0_{F} , a⟩⟩) = 0_{F}.

(33) For every field F and for all elements a, b of the support of F holds −_{F} (·_{F} (⟨⟨a, b⟩⟩)) = ·_{F} (⟨⟨a, −_{F} (b)⟩⟩).

(34) For every field F holds ·_{F} (⟨⟨1_{F} , 0_{F} ⟩⟩) = 0_{F}.

(35) For every field F holds ·_{F} (⟨⟨0_{F} , 1_{F} ⟩⟩) = 0_{F}.

(36) For every field F and for all elements a, b of the support of F holds ·_{F} (⟨⟨a, b⟩⟩) is an element of the support of F.

(37) For every field F and for all elements a, b, c of the support of F holds ·_{F} (⟨⟨·_{F} (⟨⟨a, b⟩⟩), c⟩⟩) = ·_{F} (⟨⟨a, ·_{F} (⟨⟨b, c⟩⟩)⟩⟩).

(38) For every field F and for all elements a, b of the support of F holds ·_{F} (⟨⟨a, b⟩⟩) = ·_{F} (⟨⟨b, a⟩⟩).

(39) For every field F and for every element a of the support of F holds ·_{F} (⟨⟨a, 1_{F} ⟩⟩) = a and ·_{F} (⟨⟨1_{F} , a⟩⟩) = a.

Let F be a field. The functor {−1 \atop F} yielding a function from the support of F \ single(0_{F}) into the support of F \ single(0_{F}) is defined by:

(Def.9) for every element x of the support of F \single(0_{F}) holds ·_{F} (⟨⟨x, {−1 \atop F}(x)⟩⟩) = 1_{F}.

One can prove the following propositions:

(40) For every field F and for every element x of the support of F \single(0_{F}) holds ·_{F} (⟨⟨x, {−1 \atop F}(x)⟩⟩) = 1_{F}.

(41) For every field F and for every function S from the support of F \ single(0_{F}) into the support of F \single(0_{F}) holds S = {−1 \atop F} if and only if for every element x of the support of F \ single(0_{F}) holds ·_{F} (⟨⟨x, S(x)⟩⟩) = 1_{F}.

(42) For every field F and for every element x of the support of F \ single(0_{F}) and for every element y of the support of F \single(0_{F}) such that ·_{F} (⟨⟨x, y⟩⟩) = 1_{F} holds y = {−1 \atop F}(x).

(43) For every field F and for every element x of the support of F \single(0_{F}) holds x = {−1 \atop F}({−1 \atop F}(x)).

(44) For every field F and for all elements a, b of the support of F \single(0_{F}) holds ·_{F} (⟨⟨a, b⟩⟩) is an element of the support of F \ single(0_{F}) and {−1 \atop F}(a) is an element of the support of F \ single(0_{F}).

(45) For every field F and for all elements a, b, c of the support of F such that +_{F} (⟨⟨a, b⟩⟩) = +_{F} (⟨⟨a, c⟩⟩) holds b = c.

(46) For every field F and for every element a of the support of F \single(0_{F}) and for all elements b, c of the support of F such that ·_{F} (⟨⟨a, b⟩⟩) = ·_{F} (⟨⟨a, c⟩⟩) holds b = c.

References

[1] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

Received June 20, 1990

Filters - Part I

Grzegorz Bancerek^{1} Warsaw University Białystok

Summary. Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a nonempty set of elements of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there exists an ultrafilter consists it.

MML Identifier: FILTER 0.

The articles [3], [1], [4], [7], [5], [6], and [2] provide the notation and terminology for this paper. We adopt the following convention: L is a lattice, p, p_{1}, q, q_{1}, r, r_{1} are elements of the carrier of L, and x is arbitrary. Let E be a non-empty set, and let p be an element of E. Then {p} is a non-empty subset of E.

Let E be a non-empty set, and let D_{1}, D_{2} be non-empty subsets of E. Then D_{1} ∪ D_{2} is a non-empty subset of E.

The following propositions are true:

(1) If p ⊑ q, then r ⊔ p ⊑ r ⊔ q and p ⊔ r ⊑ q ⊔ r and p ⊔ r ⊑ r ⊔ q and r ⊔ p ⊑ q ⊔ r.

(2) If p ⊑ r, then p ⊓ q ⊑ r and q ⊓ p ⊑ r.

(3) If p ⊑ r, then p ⊑ q ⊔ r and p ⊑ r ⊔ q.

(4) If p ⊑ p_{1} and q ⊑ q_{1}, then p ⊔ q ⊑ p_{1} ⊔ q_{1} and p ⊔ q ⊑ q_{1} ⊔ p_{1}.

(5) If p ⊑ p_{1} and q ⊑ q_{1}, then p ⊓ q ⊑ p_{1} ⊓ q_{1} and p ⊓ q ⊑ q_{1} ⊓ p_{1}.

(6) If p ⊑ r and q ⊑ r, then p ⊔ q ⊑ r.

(7) If r ⊑ p and r ⊑ q, then r ⊑ p ⊓ q. Let us consider L. A non-empty subset of the carrier of L is said to be a

filter of L if:

1Supported by RPBP III-24.C1.

(Def.1) p ∈ it and q ∈ it if and only if p ⊓ q ∈ it.

One can prove the following two propositions:

(8) For every non-empty subset D of the carrier of L holds D is a filter of L if and only if for all p, q holds p ∈ D and q ∈ D if and only if p ⊓ q ∈ D.

(9) For every non-empty subset D of the carrier of L holds D is a filter of L if and only if for all p, q such that p ∈ D and q ∈ D holds p ⊓ q ∈ D and for all p, q such that p ∈ D and p ⊑ q holds q ∈ D.

In the sequel H, F are filters of L. We now state several propositions:

(10) If p ∈ H, then p ⊔ q ∈ H and q ⊔ p ∈ H.

(11) There exists p such that p ∈ H.

(12) If L is an upper bound lattice, then ⊤_{L} ∈ H.

(13) If L is an upper bound lattice, then {⊤_{L}} is a filter of L.

(14) If {p} is a filter of L, then L is an upper bound lattice.

(15) The carrier of L is a filter of L.

Let us consider L. The functor [L] yields a filter of L and is defined by: (Def.2) [L] = the carrier of L.

One can prove the following proposition

(16) [L] = the carrier of L.

Let us consider L, p. The functor [p] yields a filter of L and is defined as follows: (Def.3) [p] = {q : p ⊑ q}.

One can prove the following four propositions:

(17) [p] = {q : p ⊑ q}.

(18) q ∈ [p] if and only if p ⊑ q.

(19) p ∈ [p] and p ⊔ q ∈ [p] and q ⊔ p ∈ [p].

(20) If L is a lower bound lattice, then [L] = [⊥_{L}].

Let us consider L, F. We say that F is ultrafilter if and only if: (Def.4) F ≠ the carrier of L and for every H such that F ⊆ H and H ≠ the

carrier of L holds F = H. One can prove the following four propositions:

(21) F is ultrafilter if and only if F ≠ the carrier of L and for every H such that F ⊆ H and H ≠ the carrier of L holds F = H.

(22) If L is a lower bound lattice, then for every F such that F ≠ the carrier of L there exists H such that F ⊆ H and H is ultrafilter.

(23) If there exists r such that p ⊓ r ≠ p, then [p] ≠ the carrier of L.

(24) If L is a lower bound lattice and p ≠ ⊥_{L}, then there exists H such that

p ∈ H and H is ultrafilter. In the sequel D is a non-empty subset of the carrier of L. Let us consider L,

D. The functor [D] yields a filter of L and is defined by:

(Def.5) D ⊆ [D] and for every F such that D ⊆ F holds [D] ⊆ F.

One can prove the following two propositions:

(25) D ⊆ [D] and for every F such that D ⊆ F holds [D] ⊆ F.

(26) [F] = F.

In the sequel D_{1}, D_{2} will be non-empty subsets of the carrier of L. We now state several propositions:

(27) If D_{1} ⊆ D_{2}, then [D_{1}] ⊆ [D_{2}].

(28) [[D]] ⊆ [D].

(29) If p ∈ D, then [p] ⊆ [D].

(30) If D = {p}, then [D] = [p].

(31) If L is a lower bound lattice and ⊥_{L} ∈ D, then [D] = [L] and [D] = the carrier of L.

(32) If L is a lower bound lattice and ⊥_{L} ∈ F, then F = [L] and F = the

carrier of L. Let us consider L, F. We say that F is prime if and only if:

(Def.6) p ⊔ q ∈ F if and only if p ∈ F or q ∈ F.

One can prove the following two propositions:

(33) F is prime if and only if for all p, q holds p ⊔ q ∈ F if and only if p ∈ F or q ∈ F.

(34) If L is a boolean lattice, then for all p, q holds p ⊓ (p^{c} ⊔ q) ⊑ q and for

every r such that p ⊓ r ⊑ q holds r ⊑ p^{c} ⊔ q. A lattice is called a implicative lattice if:

(Def.7) for every elements p, q of the carrier of it there exists an element r of the carrier of it such that p ⊓ r ⊑ q and for every element r_{1} of the carrier of it such that p ⊓ r_{1} ⊑ q holds r_{1} ⊑ r.

One can prove the following proposition

(35) L is a implicative lattice if and only if for every p, q there exists r such

that p ⊓ r ⊑ q and for every r_{1} such that p ⊓ r_{1} ⊑ q holds r_{1} ⊑ r. Let us consider L, p, q. Let us assume that L is a implicative lattice. The

functor p ⇒ q yields an element of the carrier of L and is defined as follows: (Def.8) p ⊓ (p ⇒ q) ⊑ q and for every r such that p ⊓ r ⊑ q holds r ⊑ p ⇒ q.

The following proposition is true

(36) If L is a implicative lattice, then for all p, q, r holds r = p ⇒ q if and

only if p ⊓ r ⊑ q and for every r_{1} such that p ⊓ r_{1} ⊑ q holds r_{1} ⊑ r. In the sequel I will denote a implicative lattice and i will denote an element

of the carrier of I. The following three propositions are true:

(37) I is an upper bound lattice.

(38) i ⇒ i = ⊤_{I}.

(39) I is a distributive lattice.

In the sequel B is a boolean lattice and F_{1}, H_{1} are filters of B. Next we state the proposition

(40) B is a implicative lattice.

We see that the implicative lattice is a distributive lattice.

For simplicity we follow the rules: I will be a implicative lattice, i, j, k will be elements of the carrier of I, D_{3} will be a non-empty subset of the carrier of I, and F_{2} will be a filter of I. The following propositions are true:

(41) If i ∈ F_{2} and i ⇒ j ∈ F_{2}, then j ∈ F_{2}.

(42) If j ∈ F_{2}, then i ⇒ j ∈ F_{2}.

Let us consider L, D_{1}, D_{2}. The functor D_{1} ⊓D_{2} yielding a non-empty subset of the carrier of L is defined as follows: (Def.9) D_{1} ⊓ D_{2} = {p ⊓ q : p ∈ D_{1} ∧ q ∈ D_{2}}.

Next we state four propositions:

(43) D_{1} ⊓ D_{2} = {p ⊓ q : p ∈ D_{1} ∧ q ∈ D_{2}}.

(44) If p ∈ D_{1} and q ∈ D_{2}, then p ⊓ q ∈ D_{1} ⊓ D_{2} and q ⊓ p ∈ D_{1} ⊓ D_{2}.

(45) If x ∈ D_{1} ⊓ D_{2}, then there exist p, q such that x = p ⊓ q and p ∈ D_{1} and q ∈ D_{2}.

(46) D_{1} ⊓ D_{2} = D_{2} ⊓ D_{1}.

Let L be a distributive lattice, and let F_{3}, F_{4} be filters of L. Then F_{3} ⊓ F_{4} is a filter of L.

Let L be a boolean lattice, and let F_{3}, F_{4} be filters of L. Then F_{3} ⊓ F_{4} is a filter of L.

One can prove the following propositions:

(47) [D_{1} ∪ D_{2}] = [[D_{1}] ∪ D_{2}] and [D_{1} ∪ D_{2}] = [D_{1} ∪ [D_{2}]].

(48) [F ∪ H] = {r : ⋁_{pq}[p ⊓ q ⊑ r ∧ p ∈ F ∧ q ∈ H]}.

(49) F ⊆ F ⊓ H and H ⊆ F ⊓ H.

(50) [F ∪ H] = [F ⊓ H].

In the sequel F_{3}, F_{4} are filters of I. The following four propositions are true:

(51) [F_{3} ∪ F_{4}] = F_{3} ⊓ F_{4}.

(52) [F_{1} ∪ H_{1}] = F_{1} ⊓ H_{1}.

(53) If j ∈ [D_{3} ∪ {i}], then i ⇒ j ∈ [D_{3}].

(54) If i ⇒ j ∈ F_{2} and j ⇒ k ∈ F_{2}, then i ⇒ k ∈ F_{2}.

In the sequel a, b, c will denote elements of the carrier of B. One can prove the following propositions:

(55) a ⇒ b = a^{c} ⊔ b.

(56) a ⊑ b if and only if a ⊓ b^{c} = ⊥_{B}.

(57) F_{1} is ultrafilter if and only if F_{1} ≠ the carrier of B and for every a holds a ∈ F_{1} or a^{c} ∈ F_{1}.

(58) F_{1} ≠ [B] and F_{1} is prime if and only if F_{1} is ultrafilter.

(59) If F_{1} is ultrafilter, then for every a holds a ∈ F_{1} if and only if a^{c} ∉ F_{1}.

(60) If a ≠ b, then there exists F_{1} such that F_{1} is ultrafilter but a ∈ F_{1} and

b ∉ F_{1} or a ∉ F_{1} and b ∈ F_{1}. In the sequel o_{1}, o_{2} are binary operations on F. Let us consider L, F. The

functor _{F} yielding a lattice is defined as follows: (Def.10) there exist o_{1}, o_{2} such that o_{1} = (the join operation of L) [: F, F :] and

o_{2} = (the meet operation of L) [: F, F :] and _{F} = ⟨F, o_{1}, o_{2}⟩. In the sequel K is a lattice. Next we state a number of propositions:

(61) K = _{F} if and only if there exist o_{1}, o_{2} such that o_{1} = (the join operation of L) [: F, F :] and o_{2} = (the meet operation of L) [: F, F :] and K = ⟨F, o_{1}, o_{2}⟩.

(62) _{[L]} = L.

(63) The carrier of _{F} = F and the join operation of _{F} = (the join operation of L) [: F, F :] and the meet operation of _{F} = (the meet operation of L) [: F, F :].

(64) For all p, q and for all elements p′, q′ of the carrier of _{F} such that p = p′ and q = q′ holds p ⊔ q = p′ ⊔ q′ and p ⊓ q = p′ ⊓ q′.

(65) For all p, q and for all elements p′, q′ of the carrier of _{F} such that p = p′ and q = q′ holds p ⊑ q if and only if p′ ⊑ q′.

(66) If L is an upper bound lattice, then _{F} is an upper bound lattice.

(67) If L is a modular lattice, then _{F} is a modular lattice.

(68) If L is a distributive lattice, then _{F} is a distributive lattice.

(69) If L is a implicative lattice, then _{F} is a implicative lattice.

(70) _{[p]} is a lower bound lattice.

(71) ⊥_{_{[p]}} = p.

(72) If L is an upper bound lattice, then ⊤_{_{[p]}} = ⊤_{L}.

(73) If L is an upper bound lattice, then _{[p]} is a bound lattice.

(74) If L is a complemented lattice and L is a modular lattice, then _{[p]} is a complemented lattice.

(75) If L is a boolean lattice, then _{[p]} is a boolean lattice.

Let us consider L, p, q. The functor p ⇔ q yielding an element of the carrier of L is defined by:

(Def.11) p ⇔ q = p ⇒ q ⊓ q ⇒ p.

Next we state three propositions:

(76) p ⇔ q = p ⇒ q ⊓ q ⇒ p.

(77) p ⇔ q = q ⇔ p.

(78) If i ⇔ j ∈ F_{2} and j ⇔ k ∈ F_{2}, then i ⇔ k ∈ F_{2}.

Let us consider L, F. The functor ≡_{F} yielding a binary relation is defined as follows:

(Def.12) field ≡_{F} ⊆ the carrier of L and for all p, q holds ⟨⟨p, q⟩⟩ ∈ ≡_{F} if and only if p ⇔ q ∈ F.

In the sequel R will denote a binary relation. We now state several propositions:

(79) R = ≡_{F} if and only if field R ⊆ the carrier of L and for all p, q holds ⟨⟨p, q⟩⟩ ∈ R if and only if p ⇔ q ∈ F.

(80) ≡_{F} is a binary relation on the carrier of L.

(81) If L is a implicative lattice, then ≡_{F} is reflexive in the carrier of L.

(82) ≡_{F} is symmetric in the carrier of L.

(83) If L is a implicative lattice, then ≡_{F} is transitive in the carrier of L.

(84) If L is a implicative lattice, then ≡_{F} is an equivalence relation of the carrier of L.

(85) If L is a implicative lattice, then field ≡_{F} = the carrier of L. Let us consider I, F_{2}. Then ≡_{F_{2}} is an equivalence relation of the carrier of

I.

Let us consider B, F_{1}. Then ≡_{F_{1}} is an equivalence relation of the carrier of B.

Let us consider L, F, p, q. The predicate p ≡_{F} q is defined by: (Def.13) p ⇔ q ∈ F.

Next we state several propositions:

(86) p ≡_{F} q if and only if p ⇔ q ∈ F.

(87) p ≡_{F} q if and only if ⟨⟨p, q⟩⟩ ∈ ≡_{F}.

(88) i ≡_{F_{2}} i and a ≡_{F_{1}} a.

(89) If p ≡_{F} q, then q ≡_{F} p.

(90) If i ≡_{F_{2}} j and j ≡_{F_{2}} k, then i ≡_{F_{2}} k but if a ≡_{F_{1}} b and b ≡_{F_{1}} c, then a ≡_{F_{1}} c.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[3] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[4] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[5] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[6] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Filters - Part I 819

[7] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathemat-

ics, 1(1):215–222, 1990. Received July 3, 1990 820

Groups

Wojciech A. Trybulec Warsaw University

Summary. Notions of group and abelian group are introduced. The power of an element of a group, order of group and order of an element of a group are defined. Basic theorems concerning those notions are presented.

MML Identifier: GROUP 1.

The notation and terminology used in this paper are introduced in the following articles: [6], [7], [9], [2], [3], [5], [12], [11], [1], [8], [4], [10], and [13]. We follow the rules: x is arbitrary, m, n are natural numbers, and i, j are integers. Let N be a non-empty subset of , and let D be a non-empty set, and let f be a function from N into D, and let n be an element of N. Then f(n) is an element of D.

Let D be a non-empty set, and let N be a non-empty subset of , and let E be a non-empty set, and let f be a function from [: D, N :] into E, and let h be an element of D, and let n be an element of N. Then f(h, n) is an element of E.

Let us consider i. Then |i| is a natural number.

We consider half group structures which are systems

⟨a carrier, an operation⟩, where the carrier is a non-empty set and the operation is a binary operation on the carrier. In the sequel S denotes a half group structure. Let us consider S. An element of S is an element of the carrier of S.

In the sequel r, s, s_{1}, s_{2}, t will be elements of S. Let us consider S, x. The predicate x ∈ S is defined as follows: (Def.1) x ∈ the carrier of S.

The following propositions are true:

(1) x ∈ S if and only if x ∈ the carrier of S.

(2) s ∈ S.

(3) If x ∈ S, then x is an element of S. Let us consider S, s_{1}, s_{2}. The functor s_{1} · s_{2} yielding an element of S is

defined by: (Def.2) s_{1} · s_{2} = (the operation of S)(s_{1}, s_{2}).

One can prove the following proposition

(4) s_{1} · s_{2} = (the operation of S)(s_{1}, s_{2}). A half group structure is called a group if:

(Def.3) (i) for all elements f, g, h of it holds (f · g) · h = f · (g · h),

(ii) there exists an element e of it such that for every element h of it holds h·e = h and e·h = h and there exists an element g of it such that h·g = e and g · h = e.

We now state three propositions:

(5) If for all r, s, t holds (r · s) · t = r · (s · t) and there exists t such that for every s_{1} holds s_{1} · t = s_{1} and t · s_{1} = s_{1} and there exists s_{2} such that s_{1} · s_{2} = t and s_{2} · s_{1} = t, then S is a group.

(6) If for all r, s, t holds (r · s) · t = r · (s · t) and for all r, s holds there exists t such that r · t = s and there exists t such that t · r = s, then S is a group.

(7) ⟨ , +_{}⟩ is a group. We follow a convention: G denotes a group and e, f, g, h denote elements of

G. Next we state two propositions: (8) (h · g) · f = h · (g · f).

(9) There exists e such that for every h holds h · e = h and e · h = h and there exists g such that h · g = e and g · h = e.

Let us consider G. The functor 1_{G} yielding an element of G is defined by: (Def.4) h · (1_{G}) = h and (1_{G}) · h = h.

One can prove the following two propositions:

(10) If for every h holds h · e = h and e · h = h, then e = 1_{G}.

(11) h · (1_{G}) = h and (1_{G}) · h = h.

Let us consider G, h. The functor h^{−1} yields an element of G and is defined as follows: (Def.5) h · (h^{−1}) = 1_{G} and (h^{−1}) · h = 1_{G}.

One can prove the following propositions:

(12) If h · g = 1_{G} and g · h = 1_{G}, then g = h^{−1}.

(13) h · h^{−1} = 1_{G} and h^{−1} · h = 1_{G}.

(14) If h · g = h · f or g · h = f · h, then g = f.

(15) If h · g = h or g · h = h, then g = 1_{G}.

(16) (1_{G})^{−1} = 1_{G}.

(17) If h^{−1} = g^{−1}, then h = g.

(18) If h^{−1} = 1_{G}, then h = 1_{G}.

(19) (h^{−1})^{−1} = h.

(20) If h · g = 1_{G} or g · h = 1_{G}, then h = g^{−1} and g = h^{−1}.

(21) h · f = g if and only if f = h^{−1} · g.

(22) f · h = g if and only if f = g · h^{−1}.

(23) There exists f such that g · f = h.

(24) There exists f such that f · g = h.

(25) (h · g)^{−1} = g^{−1} · h^{−1}.

(26) g · h = h · g if and only if (g · h)^{−1} = g^{−1} · h^{−1}.

(27) g · h = h · g if and only if g^{−1} · h^{−1} = h^{−1} · g^{−1}.

(28) g · h = h · g if and only if g · h^{−1} = h^{−1} · g.

In the sequel u is a unary operation on the carrier of G. Let us consider G. The functor ·_{G}^{−1} yields a unary operation on the carrier of G and is defined by: (Def.6) ·_{G}^{−1}(h) = h^{−1}.

We now state several propositions:

(29) If for every h holds u(h) = h^{−1}, then u = ·_{G}^{−1}.

(30) ·_{G}^{−1}(h) = h^{−1}.

(31) The operation of G is associative.

(32) 1_{G} is a unity w.r.t. the operation of G.

(33) 1_{the operation of G} = 1_{G}.

(34) The operation of G has a unity.

(35) ·_{G}^{−1} is an inverse operation w.r.t. the operation of G.

(36) The operation of G has an inverse operation.

(37) The inverse operation w.r.t. (the operation of G) = ·_{G}^{−1}.

Let us consider G. The functor power_{G} yields a function from [: the carrier of G, :] into the carrier of G and is defined by: (Def.7) power_{G}(h, 0) = 1_{G} and for every n holds power_{G}(h, n+1) = power_{G}(h,

n) · h. In the sequel H is a function from [: the carrier of G, :] into the carrier of

G. We now state three propositions:

(38) If for every h holds H(h, 0) = 1_{G} and for every n holds H(h, n + 1) = H(h, n) · h, then H = power_{G}.

(39) power_{G}(h, 0) = 1_{G}.

(40) power_{G}(h, n + 1) = power_{G}(h, n) · h.

Let us consider G, n, h. The functor h^{n} yields an element of G and is defined as follows: (Def.8) h^{n} = power_{G}(h, n).

We now state a number of propositions:

(41) h^{n} = power_{G}(h, n).

(42) (1_{G})^{n} = 1_{G}.

(43) h^{0} = 1_{G}.

(44) h^{1} = h.

(45) h^{2} = h · h.

(46) h^{3} = (h · h) · h.

(47) h^{2} = 1_{G} if and only if h^{−1} = h.

(48) h^{n+m} = h^{n} · h^{m} and h^{m+n} = h^{n} · h^{m}.

(49) h^{n+1} = h^{n} · h and h^{n+1} = h · h^{n} and h^{1+n} = h^{n} · h and h^{1+n} = h · h^{n}.

(50) h^{n·m} = (h^{n})^{m}.

(51) (h^{−1})^{n} = (h^{n})^{−1}.

(52) If g · h = h · g, then g · h^{n} = h^{n} · g.

(53) If g · h = h · g, then g^{n} · h^{m} = h^{m} · g^{n}.

(54) If g · h = h · g, then (g · h)^{n} = g^{n} · h^{n}.

Let us consider G, i, h. The functor h^{i} yielding an element of G is defined by: (Def.9) h^{i} = h^{|i|} if 0 ≤ i, h^{i} = (h^{|i|})^{−1}, otherwise.

The following propositions are true:

(55) If 0 ≤ i, then h^{i} = h^{|i|}.

(56) If 0 ≰ i, then h^{i} = (h^{|i|})^{−1}.

(57) If i < 0, then h^{i} = (h^{|i|})^{−1}.

(58) If i = n, then h^{i} = h^{n}.

(59) If i = 0, then h^{i} = 1_{G}.

(60) If i ≤ 0, then h^{i} = (h^{|i|})^{−1}.

(61) (1_{G})^{i} = 1_{G}.

(62) h^{−1} = h^{−1}.

(63) h^{i+j} = h^{i} · h^{j}.

(64) h^{n+j} = h^{n} · h^{j}.

(65) h^{i+m} = h^{i} · h^{m}.

(66) h^{j+1} = h^{j} · h and h^{j+1} = h · h^{j} and h^{1+j} = h^{j} · h and h^{1+j} = h · h^{j}.

(67) h^{i·j} = (h^{i})^{j}.

(68) h^{n·j} = (h^{n})^{j}.

(69) h^{i·m} = (h^{i})^{m}.

(70) h^{−i} = (h^{i})^{−1}.

(71) h^{−n} = (h^{n})^{−1}.

(72) (h^{−1})^{i} = (h^{i})^{−1}.

(73) If g · h = h · g, then (g · h)^{i} = g^{i} · h^{i}.

(74) If g · h = h · g, then g^{i} · h^{j} = h^{j} · g^{i}.

(75) If g · h = h · g, then g^{n} · h^{j} = h^{j} · g^{n}.

(76) If g · h = h · g, then g^{i} · h^{m} = h^{m} · g^{i}.

(77) If g · h = h · g, then g · h^{i} = h^{i} · g.

Let us consider G, h. We say that h is of order 0 if and only if: (Def.10) if h^{n} = 1_{G}, then n = 0.

We now state two propositions:

(78) h is of order 0 if and only if for every n such that h^{n} = 1_{G} holds n = 0.

(79) 1_{G} is not of order 0.

Let us consider G, h. The functor ord(h) yields a natural number and is defined by:

(Def.11) ord(h) = 0 if h is of order 0, h^{ord(h)} = 1_{G} and ord(h) ≠ 0 and for every

m such that h^{m} = 1_{G} and m ≠ 0 holds ord(h) ≤ m, otherwise. One can prove the following propositions:

(80) If h is not of order 0 and h^{m} = 1_{G} and m ≠ 0 and for every n such that h^{n} = 1_{G} and n ≠ 0 holds m ≤ n, then m = ord(h).

(81) h is of order 0 if and only if ord(h) = 0.

(82) h^{ord(h)} = 1_{G}.

(83) If h is not of order 0 and h^{m} = 1_{G} and m ≠ 0, then ord(h) ≤ m.

(84) ord(1_{G}) = 1.

(85) If ord(h) = 1, then h = 1_{G}.

(86) If h^{n} = 1_{G}, then ord(h) | n.

Let us consider G. The functor Ord(G) yielding a cardinal number is defined as follows:

(Def.12) Ord(G) = \overline{\overline{the carrier of G}}.

We now state the proposition

(87) Ord(G) = \overline{\overline{the carrier of G}}.

We now define two new predicates. Let us consider G. We say that G is finite if and only if:

(Def.13) the carrier of G is finite.

We say that G is infinite if and only if G is not finite.

The following proposition is true

(88) G is finite if and only if the carrier of G is finite.

Let us consider G. Let us assume that G is finite. The functor ord(G) yielding a natural number is defined by:

(Def.14) ord(G) = card (the carrier of G).

Next we state two propositions:

(89) If G is finite, then ord(G) = card (the carrier of G).

(90) If G is finite, then ord(G) ≥ 1.

A group is called an Abelian group if: (Def.15) for all elements a, b of it holds a · b = b · a.

We now state two propositions:

(91) If for all h, g holds h · g = g · h, then G is an Abelian group.

(92) ⟨ , +_{}⟩ is an Abelian group.

In the sequel A is an Abelian group and a, b are elements of A. One can prove the following propositions:

(93) a · b = b · a.

(94) (a · b)^{−1} = a^{−1} · b^{−1}.

(95) (a · b)^{n} = a^{n} · b^{n}.

(96) (a · b)^{i} = a^{i} · b^{i}.

(97) ⟨The carrier of A, the operation of A, ·_{A}^{−1}, 1_{A}⟩ is an Abelian group. In the sequel B denotes an Abelian group. We now state two propositions:

(98) ⟨The carrier of B, the addition of B⟩ is an Abelian group.

(99) −1 < 0.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175– 180, 1990.

[5] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[9] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[10] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[11] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

[12] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

[13] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

828

The Divisibility of Integers and Integer Relatively Primes ^{1}

Rafał Kwiatek Nicolaus Copernicus University Toruń Grzegorz Zwara Warsaw University Białystok

Summary. We introduce the following notions: 1)the least common multiple of two integers (lcm(i, j)), 2)the greatest common divisor of two integers (gcd(i, j)), 3)the relative prime integer numbers, 4)the prime numbers. A few facts concerning the above items, among them a so-called Foundamental Theorem of Arithmetic, are introduced.

MML Identifier: INT 2.

The papers [2], [1], and [3] provide the terminology and notation for this paper. In the sequel a, b will be natural numbers. Next we state several propositions:

(1) lcm(a, b) = lcm(b, a).

(2) gcd(a, b) = gcd(b, a).

(3) 0 | a if and only if a = 0.

(4) a = 0 or b = 0 if and only if lcm(a, b) = 0.

(5) a = 0 and b = 0 if and only if gcd(a, b) = 0.

(6) a · b = lcm(a, b) · gcd(a, b). We follow the rules: m, n are natural numbers and a, b, c, a_{1}, b_{1} are integers.

Let us consider n. The functor +n yields an integer and is defined by: (Def.1) +n = n.

Next we state a number of propositions:

(7) +n = n.

(8) −n is a natural number if and only if n = 0.

(9) −1 is not a natural number.

(10) +0 | a if and only if a = 0.

(11) a | a and a | −a and −a | a.

1Supported by RPBP.III-24.B5.

(12) If a | b, then a | b · c.

(13) If a | b and b | c, then a | c.

(14) a | b if and only if a | −b but a | b if and only if −a | b but a | b if and only if −a | −b but a | −b if and only if −a | b.

(15) If a | b and b | a, then a = b or a = −b.

(16) a | +0 and +1 | a and −1 | a.

(17) If a | +1 or a | −1, then a = 1 or a = −1.

(18) If a = 1 or a = −1, then a | +1 and a | −1.

(19) a ≡ b(mod c) if and only if c | a − b.

(20) |a| is a natural number.

Let us consider a. Then |a| is a natural number.

We now state the proposition

(21) a | b if and only if |a| | |b|.

Let us consider a, b. The functor lcm(a, b) yields an integer and is defined as follows: (Def.2) lcm(a, b) = lcm(|a|, |b|).

The following propositions are true:

(22) lcm(a, b) = lcm(|a|, |b|).

(23) lcm(a, b) is a natural number.

(24) lcm(a, b) = lcm(b, a).

(25) a | lcm(a, b).

(26) b | lcm(a, b).

(27) For every c such that a | c and b | c holds lcm(a, b) | c.

Let us consider a, b. The functor gcd(a, b) yields an integer and is defined by: (Def.3) gcd(a, b) = gcd(|a|, |b|).

One can prove the following propositions:

(28) gcd(a, b) = gcd(|a|, |b|).

(29) gcd(a, b) is a natural number.

(30) gcd(a, b) = gcd(b, a).

(31) gcd(a, b) | a.

(32) gcd(a, b) | b.

(33) For every c such that c | a and c | b holds c | gcd(a, b).

(34) a = 0 or b = 0 if and only if lcm(a, b) = 0.

(35) a = 0 and b = 0 if and only if gcd(a, b) = 0.

Let us consider a, b. We say that a and b are relatively prime if and only if: (Def.4) gcd(a, b) = 1.

Next we state several propositions:

(36) a and b are relatively prime if and only if gcd(a, b) = 1.

(37) If a and b are relatively prime, then b and a are relatively prime.

(38) If a ≠ 0 or b ≠ 0, then there exist a_{1}, b_{1} such that a = gcd(a, b) · a_{1} and b = gcd(a, b) · b_{1} and a_{1} and b_{1} are relatively prime.

(39) If a and b are relatively prime, then gcd(c·a, c·b) = |c| and gcd(c·a, b·c) = |c| and gcd(a · c, c · b) = |c| and gcd(a · c, b · c) = |c|.

(40) If c | a · b and a and c are relatively prime, then c | b.

(41) If a and c are relatively prime and b and c are relatively prime, then

a · b and c are relatively prime. In the sequel p, q, k, l will denote natural numbers. Let us consider p. We

say that p is prime if and only if: (Def.5) p > 1 and for every n such that n | p holds n = 1 or n = p.

The following proposition is true

(42) p is prime if and only if p > 1 and for every n such that n | p holds

n = 1 or n = p. Let us consider m, n. We say that m and n are relatively prime if and only

if: (Def.6) gcd(m, n) = 1.

We now state several propositions:

(43) m and n are relatively prime if and only if gcd(m, n) = 1.

(44) 2 is prime.

(45) There exists p such that p is prime.

(46) There exists p such that p is not prime.

(47) If p is prime and q is prime, then p and q are relatively prime or p = q. In this article we present several logical schemes. The scheme Ind1 concerns

a natural number A, and a unary predicate P, and states that:

for every k such that k ≥ A holds P[k] provided the parameters meet the following conditions:

• P[A],

• for every k such that k ≥ A and P[k] holds P[k + 1].

The scheme Comp Ind1 concerns a natural number A, and a unary predicate P, and states that:

for every k such that k ≥ A holds P[k] provided the parameters have the following property:

• for every k such that k ≥ A and for every n such that n ≥ A and n < k holds P[n] holds P[k].

Next we state the proposition

(48) If l ≥ 2, then there exists p such that p is prime and p | l.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[3] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

From Loops to Abelian Multiplicative Groups with Zero ^{1}

Michał Muzalewski Warsaw University

Białystok

Wojciech Skaba Nicolaus Copernicus University Toruń

Summary. Elementary axioms and theorems on the theory of algebraic structures, taken from the book [4]. First a loop structure ⟨G, 0, +⟩ is defined and six axioms corresponding to it are given. Group is defined by extending the set of axioms with (a+b)+c = a+(b+c). At the same time an alternate approach to the set of axioms is shown and both sets are proved to yield the same algebraic structure. A trivial example of loop is used to ensure the existence of the modes being constructed. A multiplicative group is contemplated, which is quite similar to the previously defined additive group (called simply a group here), but is supposed to be of greater interest in the future considerations of algebraic structures. The final section brings a slightly more sophisticated structure i.e: a multiplicative loop/group with zero: ⟨G, ·, 1, 0⟩. Here the proofs are a more challenging and the above trivial example is replaced by a more common (and comprehensive) structure built on the foundation of real numbers.

MML Identifier: ALGSTR 1.

The notation and terminology used in this paper are introduced in the following articles: [1], [2], and [3]. We consider loop structures which are systems

⟨a carrier, an addition, a zero⟩, where the carrier is a non-empty set, the addition is a binary operation on the carrier, and the zero is an element of the carrier. In the sequel G_{1} will denote a loop structure. Let us consider G_{1}. An element of G_{1} is an element of the carrier of G_{1}.

In the sequel a, b will denote elements of G_{1}. Let us consider G_{1}, a, b. The functor a + b yielding an element of G_{1} is defined as follows: (Def.1) a + b = (the addition of G_{1})(a, b).

We now state the proposition

1Supported by RPBP.III-24.C6.

(1) a + b = (the addition of G_{1})(a, b). Let us consider G_{1}. The functor 0_{G_{1}} yielding an element of G_{1} is defined as

follows: (Def.2) 0_{G_{1}} = the zero of G_{1}.

One can prove the following proposition

(2) 0_{G_{1}} = the zero of G_{1}. Let x be arbitrary. The functor Extract(x) yielding an element of {x} is

defined by: (Def.3) Extract(x) = x.

One can prove the following proposition

(3) For an arbitrary x holds Extract(x) = x. The trivial loop a loop structure is defined as follows:

(Def.4) the trivial loop = ⟨{0}, zo, Extract(0)⟩.

One can prove the following three propositions:

(4) The trivial loop = ⟨{0}, zo, Extract(0)⟩.

(5) If a is an element of the trivial loop, then a = 0_{the trivial loop}.

(6) For all elements a, b of the trivial loop holds a + b = 0_{the trivial loop}. A loop structure is called a loop if:

(Def.5) (i) for every element a of it holds a + 0_{it} = a, (ii) for every element a of it holds 0_{it} + a = a, (iii) for every elements a, b of it there exists an element x of it such that

a + x = b, (iv) for every elements a, b of it there exists an element x of it such that

x + a = b,

(v) for all elements a, x, y of it such that a + x = a + y holds x = y, (vi) for all elements a, x, y of it such that x + a = y + a holds x = y.

The following proposition is true

(7) Let G_{1} be a loop structure. Then G_{1} is a loop if and only if the following conditions are satisfied:

(i) for every element a of G_{1} holds a + 0_{G_{1}} = a, (ii) for every element a of G_{1} holds 0_{G_{1}} + a = a, (iii) for every elements a, b of G_{1} there exists an element x of G_{1} such that

a + x = b, (iv) for every elements a, b of G_{1} there exists an element x of G_{1} such that

x + a = b,

(v) for all elements a, x, y of G_{1} such that a + x = a + y holds x = y, (vi) for all elements a, x, y of G_{1} such that x + a = y + a holds x = y. Let us note that it makes sense to consider the following constant. Then

the trivial loop is a loop.

A loop is called a group if: (Def.6) for all elements a, b, c of it holds (a + b) + c = a + (b + c).

We now state the proposition

(8) For every loop G_{1} holds G_{1} is a group if and only if for all elements a, b, c of G_{1} holds (a + b) + c = a + (b + c).

We follow the rules: L will be a loop structure and a, b, c, x will be elements of L. We now state the proposition

(9) L is a group if and only if for every a holds a + 0_{L} = a and for every a there exists x such that a + x = 0_{L} and for all a, b, c holds (a + b) + c = a + (b + c).

Let us note that it makes sense to consider the following constant. Then the trivial loop is a group.

A group is called an Abelian group if: (Def.7) for all elements a, b of it holds a + b = b + a.

Next we state two propositions:

(10) For every group G holds G is an Abelian group if and only if for all elements a, b of G holds a + b = b + a.

(11) L is an Abelian group if and only if the following conditions are satisfied: (i) for every a holds a + 0_{L} = a, (ii) for every a there exists x such that a + x = 0_{L}, (iii) for all a, b, c holds (a + b) + c = a + (b + c), (iv) for all a, b holds a + b = b + a. Let L be a group, and let a be an element of L. The functor −a yielding an

element of L is defined by: (Def.8) a + (−a) = 0_{L}.

We now state the proposition

(12) For every group L and for every element a of L holds a + (−a) = 0_{L}. In the sequel G will denote a group and a, b will denote elements of G. One

can prove the following proposition

(13) a + (−a) = 0_{G} and (−a) + a = 0_{G}.

Let us consider G, a, b. The functor a − b yields an element of G and is defined as follows: (Def.9) a − b = a + (−b).

Next we state the proposition

(14) a − b = a + (−b).

We consider mutiplicative loop structures which are systems ⟨a carrier, a multiplication, a unity⟩,

where the carrier is a non-empty set, the multiplication is a binary operation on the carrier, and the unity is an element of the carrier. In the sequel G_{1} is a mutiplicative loop structure. Let us consider G_{1}. An element of G_{1} is an element of the carrier of G_{1}.

In the sequel a, b are elements of G_{1}. Let us consider G_{1}, a, b. The functor a · b yields an element of G_{1} and is defined as follows:

(Def.10) a · b = (the multiplication of G_{1})(a, b).

One can prove the following proposition

(15) a · b = (the multiplication of G_{1})(a, b).

Let us consider G_{1}. The functor 1_{G_{1}} yields an element of G_{1} and is defined by:

(Def.11) 1_{G_{1}} = the unity of G_{1}.

One can prove the following proposition

(16) 1_{G_{1}} = the unity of G_{1}.

The trivial multiplicative loop a mutiplicative loop structure is defined as follows:

(Def.12) the trivial multiplicative loop = ⟨{0}, zo, Extract(0)⟩.

The following propositions are true:

(17) The trivial multiplicative loop = ⟨{0}, zo, Extract(0)⟩.

(18) If a is an element of the trivial multiplicative loop, then a = 1_{the trivial multiplicative loop}.

(19) For all elements a, b of the trivial multiplicative loop holds a · b =

1_{the trivial multiplicative loop}. A mutiplicative loop structure is said to be a multiplicative loop if:

(Def.13) (i) for every element a of it holds a · (1_{it}) = a, (ii) for every element a of it holds (1_{it}) · a = a, (iii) for every elements a, b of it there exists an element x of it such that

a · x = b, (iv) for every elements a, b of it there exists an element x of it such that

x · a = b,

(v) for all elements a, x, y of it such that a · x = a · y holds x = y, (vi) for all elements a, x, y of it such that x · a = y · a holds x = y.

We now state the proposition

(20) Let L be a mutiplicative loop structure. Then L is a multiplicative loop

if and only if the following conditions are satisfied:

(i) for every element a of L holds a · (1_{L}) = a, (ii) for every element a of L holds (1_{L}) · a = a, (iii) for every elements a, b of L there exists an element x of L such that

a · x = b, (iv) for every elements a, b of L there exists an element x of L such that

x · a = b,

(v) for all elements a, x, y of L such that a · x = a · y holds x = y, (vi) for all elements a, x, y of L such that x · a = y · a holds x = y. Let us note that it makes sense to consider the following constant. Then

the trivial multiplicative loop is a multiplicative loop.

A multiplicative loop is said to be a multiplicative group if: (Def.14) for all elements a, b, c of it holds (a · b) · c = a · (b · c).

One can prove the following proposition

(21) For every multiplicative loop L holds L is a multiplicative group if and

only if for all elements a, b, c of L holds (a · b) · c = a · (b · c). We follow the rules: L is a mutiplicative loop structure and a, b, c, x are

elements of L. One can prove the following proposition

(22) L is a multiplicative group if and only if for every a holds a · (1_{L}) = a and for every a there exists x such that a · x = 1_{L} and for all a, b, c holds (a · b) · c = a · (b · c).

Let us note that it makes sense to consider the following constant. Then the trivial multiplicative loop is a multiplicative group.

A multiplicative group is called a multiplicative Abelian group if: (Def.15) for all elements a, b of it holds a · b = b · a.

The following propositions are true:

(23) For every multiplicative group G holds G is a multiplicative Abelian group if and only if for all elements a, b of G holds a · b = b · a.

(24) L is a multiplicative Abelian group if and only if the following conditions

are satisfied:

(i) for every a holds a · (1_{L}) = a, (ii) for every a there exists x such that a · x = 1_{L}, (iii) for all a, b, c holds (a · b) · c = a · (b · c), (iv) for all a, b holds a · b = b · a. Let L be a multiplicative group, and let a be an element of L. The functor

a^{−1} yields an element of L and is defined by: (Def.16) a · (a^{−1}) = 1_{L}.

The following proposition is true

(25) For every multiplicative group L and for every element a of L holds

a · a^{−1} = 1_{L}. In the sequel G is a multiplicative group and a, b are elements of G. The

following proposition is true

(26) a · a^{−1} = 1_{G} and a^{−1} · a = 1_{G}.

Let us consider G, a, b. The functor \frac{a}{b} yields an element of G and is defined by:

(Def.17) \frac{a}{b} = a · b^{−1}.

One can prove the following proposition

(27) \frac{a}{b} = a · b^{−1}.

We consider mutiplicative loop with zero structures which are systems ⟨a carrier, a multiplication, a unity, a zero⟩,

where the carrier is a non-empty set, the multiplication is a binary operation on the carrier, the unity is an element of the carrier, and the zero is an element of the carrier. In the sequel G_{1} will be a mutiplicative loop with zero structure. Let us consider G_{1}. An element of G_{1} is an element of the carrier of G_{1}.

In the sequel a, b will denote elements of G_{1}. Let us consider G_{1}, a, b. The functor a · b yielding an element of G_{1} is defined by:

(Def.18) a · b = (the multiplication of G_{1})(a, b).

The following proposition is true

(28) a · b = (the multiplication of G_{1})(a, b).

Let us consider G_{1}. The functor 1_{G_{1}} yields an element of G_{1} and is defined as follows:

(Def.19) 1_{G_{1}} = the unity of G_{1}.

One can prove the following proposition

(29) 1_{G_{1}} = the unity of G_{1}.

Let us consider G_{1}. The functor 0_{G_{1}} yielding an element of G_{1} is defined as follows:

(Def.20) 0_{G_{1}} = the zero of G_{1}.

One can prove the following proposition

(30) 0_{G_{1}} = the zero of G_{1}.

The trivial multiplicative loop_{0} a mutiplicative loop with zero structure is defined by:

(Def.21) the trivial multiplicative loop_{0} = ⟨ , ·_{}, 1, 0⟩.

One can prove the following three propositions:

(31) The trivial multiplicative loop_{0} = ⟨ , ·_{}, 1, 0⟩.

(32) For all real numbers q, p such that q ≠ 0 there exists a real number y such that p = q · y.

(33) For all real numbers q, p such that q ≠ 0 there exists a real number y

such that p = y · q. A mutiplicative loop with zero structure is called a multiplicative loop with

zero if: (Def.22) (i) 0_{it} ≠ 1_{it},

(ii) for every element a of it holds a · (1_{it}) = a, (iii) for every element a of it holds (1_{it}) · a = a, (iv) for all elements a, b of it such that a ≠ 0_{it} there exists an element x of

it such that a · x = b,

(v) for all elements a, b of it such that a ≠ 0_{it} there exists an element x of it such that x · a = b,

(vi) for all elements a, x, y of it such that a ≠ 0_{it} holds if a · x = a · y, then

x = y, (vii) for all elements a, x, y of it such that a ≠ 0_{it} holds if x · a = y · a, then

x = y, (viii) for every element a of it holds a · 0_{it} = 0_{it}, (ix) for every element a of it holds 0_{it} · a = 0_{it}.

The following proposition is true (34) Let L be a mutiplicative loop with zero structure. Then L is a multi-

plicative loop with zero if and only if the following conditions are satisfied:

(i) 0_{L} ≠ 1_{L}, (ii) for every element a of L holds a · (1_{L}) = a, (iii) for every element a of L holds (1_{L}) · a = a, (iv) for all elements a, b of L such that a ≠ 0_{L} there exists an element x of

L such that a · x = b,

(v) for all elements a, b of L such that a ≠ 0_{L} there exists an element x of L such that x · a = b,

(vi) for all elements a, x, y of L such that a ≠ 0_{L} holds if a · x = a · y, then

x = y, (vii) for all elements a, x, y of L such that a ≠ 0_{L} holds if x · a = y · a, then

x = y, (viii) for every element a of L holds a · 0_{L} = 0_{L}, (ix) for every element a of L holds 0_{L} · a = 0_{L}. Let us note that it makes sense to consider the following constant. Then

the trivial multiplicative loop_{0} is a multiplicative loop with zero.

A multiplicative loop with zero is called a multiplicative group with zero if: (Def.23) for all elements a, b, c of it holds (a · b) · c = a · (b · c).

One can prove the following proposition

(35) For every multiplicative loop L with zero holds L is a multiplicative group with zero if and only if for all elements a, b, c of L holds (a · b) · c = a · (b · c).

We follow a convention: L denotes a mutiplicative loop with zero structure and a, b, c, x denote elements of L. One can prove the following proposition (36) L is a multiplicative group with zero if and only if the following condi-

tions are satisfied:

(i) 0_{L} ≠ 1_{L}, (ii) for every a holds a · (1_{L}) = a, (iii) for every a such that a ≠ 0_{L} there exists x such that a · x = 1_{L}, (iv) for all a, b, c holds (a · b) · c = a · (b · c),

(v) for every a holds a · 0_{L} = 0_{L}, (vi) for every a holds 0_{L} · a = 0_{L}. Let us note that it makes sense to consider the following constant. Then

the trivial multiplicative loop_{0} is a multiplicative group with zero.

A multiplicative group with zero is said to be a multiplicative commutative group with zero if:

(Def.24) for all elements a, b of it holds a · b = b · a.

We now state two propositions:

(37) For every multiplicative group L with zero holds L is a multiplicative commutative group with zero if and only if for all elements a, b of L holds a · b = b · a.

(38) L is a multiplicative commutative group with zero if and only if the

following conditions are satisfied:

(i) 0_{L} ≠ 1_{L}, (ii) for every a holds a · (1_{L}) = a, (iii) for every a such that a ≠ 0_{L} there exists x such that a · x = 1_{L}, (iv) for all a, b, c holds (a · b) · c = a · (b · c),

(v) for every a holds a · 0_{L} = 0_{L}, (vi) for every a holds 0_{L} · a = 0_{L}, (vii) for all a, b holds a · b = b · a. Let L be a multiplicative group with zero, and let a be an element of L. Let

us assume that a ≠ 0_{L}. The functor a^{−1} yielding an element of L is defined as follows:

(Def.25) a · (a^{−1}) = 1_{L}.

We now state the proposition

(39) For every multiplicative group L with zero and for every element a of

L such that a ≠ 0_{L} holds a · a^{−1} = 1_{L}. In the sequel G will be a multiplicative group with zero and a, b will be

elements of G. One can prove the following proposition

(40) If a ≠ 0_{G}, then a · a^{−1} = 1_{G} and a^{−1} · a = 1_{G}.

Let us consider G, a, b. Let us assume that b ≠ 0_{G}. The functor \frac{a}{b} yields an element of G and is defined by:

(Def.26) \frac{a}{b} = a · b^{−1}.

We now state the proposition

(41) If b ≠ 0_{G}, then \frac{a}{b} = a · b^{−1}.

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175– 180, 1990.

[2] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[3] Michał Muzalewski. Midpoint algebras. Formalized Mathematics, 1(3):483– 488, 1990.

[4] Wanda Szmielew. From Affine to Euclidean Geometry. Volume 27, PWN – D.Reidel Publ. Co., Warszawa – Dordrecht, 1983.

Basic Properties of Rational Numbers

Andrzej Kondracki^{1} Warsaw University

Summary. A definition of rational numbers and some basic properties of them. Operations of addition, substraction, multiplication are redefined for rational numbers. Functors numerator (num p) and denominator (den p) (p is rational) are defined and some properties of them are presented. Density of rational numbers is also given.

MML Identifier: RAT 1.

The notation and terminology used here are introduced in the following papers: [4], [2], [1], [3], and [5]. For simplicity we follow the rules: x is arbitrary, a, b are real numbers, k, k_{1}, l, l_{1} are natural numbers, m, m_{1}, n, n_{1} are integers, and D is a non-empty set. Let us consider m. Then |m| is a natural number.

Let us consider k. Then |k| is a natural number. The non-empty set is defined by:

(Def.1) x ∈ if and only if there exist m, n such that n ≠ 0 and x = \frac{m}{n}. One can prove the following proposition

(1) D = if and only if for every x holds x ∈ D if and only if there exist m, n such that n ≠ 0 and x = \frac{m}{n}.

A real number is said to be a rational number if: (Def.2) it is an element of .

We now state a number of propositions:

(2) For every real number x holds x is a rational number if and only if x is an element of .

(4)^{2} If x ∈ , then x ∈ .

(5) x is a rational number if and only if x ∈ .

(6) x is a rational number if and only if there exist m, n such that n ≠ 0 and x = \frac{m}{n}.

1Supported by RPBP.III-24.C1.

2The proposition (3) became obvious.

(7) For every integer x holds x is a rational number.

(8) For every natural number x holds x is a rational number.

(9) 1 is a rational number and 0 is a rational number.

(10) ⊆ .

(11) ⊆ .

(12) ⊆ .

In the sequel p, q denote rational numbers. Next we state three propositions:

(13) If x = \frac{k}{l} and l ≠ 0, then x is a rational number.

(14) If x = \frac{m}{k} and k ≠ 0, then x is a rational number.

(15) If x = \frac{k}{m} and m ≠ 0, then x is a rational number.

Let us consider p, q. Then p · q is a rational number. Then p + q is a rational number. Then p − q is a rational number.

Let us consider p, m. Then p + m is a rational number. Then p − m is a rational number. Then p · m is a rational number.

Let us consider m, p. Then m + p is a rational number. Then m − p is a rational number. Then m · p is a rational number.

Let us consider p, k. Then p+k is a rational number. Then p−k is a rational number. Then p · k is a rational number.

Let us consider k, p. Then k+p is a rational number. Then k−p is a rational number. Then k · p is a rational number.

Let us consider p. Then −p is a rational number. Then |p| is a rational number.

One can prove the following propositions:

(16) For all p, q such that q ≠ 0 holds \frac{p}{q} is a rational number.

(17) If k ≠ 0, then \frac{p}{k} is a rational number.

(18) If m ≠ 0, then \frac{p}{m} is a rational number.

(19) If p ≠ 0, then \frac{k}{p} is a rational number and \frac{m}{p} is a rational number.

(20) For every p such that p ≠ 0 holds \frac{1}{p} is a rational number.

(21) For every p such that p ≠ 0 holds p^{−1} is a rational number.

(22) For all a, b such that a < b there exists p such that a < p and p < b.

(23) a < b if and only if there exists p such that a < p and p < b.

(24) For every p there exist m, k such that k ≠ 0 and p = \frac{m}{k}.

(25) For every p there exist m, k such that k ≠ 0 and p = \frac{m}{k} and for all n, l

such that l ≠ 0 and p = \frac{n}{l} holds k ≤ l. Let us consider p. The functor den p yielding a natural number is defined by:

(Def.3) den p ≠ 0 and there exists m such that p = \frac{m}{den p} and for all n, k such

that k ≠ 0 and p = \frac{n}{k} holds den p ≤ k. We now state the proposition (26) den p ≠ 0 and there exists m such that p = \frac{m}{den p} and for all n, k such

that k ≠ 0 and p = \frac{n}{k} holds den p ≤ k. Let us consider p. The functor num p yields an integer and is defined by:

(Def.4) num p = den p · p.

One can prove the following propositions:

(27) 0 < den p.

(28) 0 ≠ den p.

(29) 1 ≤ den p.

(30) 0 < den p^{−1}.

(31) 0 ≤ den p.

(32) 0 ≤ den p^{−1}.

(33) 0 ≠ den p^{−1}.

(34) 1 ≥ den p^{−1}.

(35) num p = den p · p and num p = p · den p.

(36) num p = 0 if and only if p = 0.

(37) p = \frac{num p}{den p} and p = num p · den p^{−1} and p = den p^{−1} · num p.

(38) If p ≠ 0, then den p = \frac{num p}{p}.

(39) If p = \frac{m}{k} and k ≠ 0, then den p ≤ k.

(40) If p is an integer, then den p = 1 and num p = p.

(41) If num p = p or den p = 1, then p is an integer.

(42) num p = p if and only if den p = 1.

(43) If p is a natural number, then den p = 1 and num p = p.

(44) If num p = p or den p = 1 but 0 ≤ p, then p is a natural number.

(45) 1 < den p if and only if p is not an integer.

(46) 1 > den p^{−1} if and only if p is not an integer.

(47) num p = den p if and only if p = 1.

(48) num p = − den p if and only if p = −1.

(49) − num p = den p if and only if p = −1.

(50) Suppose m ≠ 0. Then p = \frac{num p·m}{den p·m} and p = \frac{m·num p}{den p·m} and p = \frac{m·num p}{m·den p} and p = \frac{num p·m}{m·den p}.

(51) Suppose k ≠ 0. Then p = \frac{num p·k}{den p·k} and p = \frac{k·num p}{den p·k} and p = \frac{k·num p}{k·den p} and p = \frac{num p·k}{k·den p}.

(52) Suppose p = \frac{m}{n} and n ≠ 0 and m_{1} ≠ 0. Then p = \frac{m·m_{1}}{n·m_{1}} and p = \frac{m_{1}·m}{n·m_{1}} and p = \frac{m_{1}·m}{m_{1}·n} and p = \frac{m·m_{1}}{m_{1}·n}.

(53) Suppose p = \frac{m}{l} and l ≠ 0 and m_{1} ≠ 0. Then p = \frac{m·m_{1}}{l·m_{1}} and p = \frac{m_{1}·m}{l·m_{1}} and p = \frac{m_{1}·m}{m_{1}·l} and p = \frac{m·m_{1}}{m_{1}·l}.

(54) Suppose p = \frac{l}{n} and n ≠ 0 and m_{1} ≠ 0. Then p = \frac{l·m_{1}}{n·m_{1}} and p = \frac{m_{1}·l}{n·m_{1}} and p = \frac{m_{1}·l}{m_{1}·n} and p = \frac{l·m_{1}}{m_{1}·n}.

(55) Suppose p = \frac{l}{l_{1}} and l_{1} ≠ 0 and m_{1} ≠ 0. Then p = \frac{l·m_{1}}{l_{1}·m_{1}} and p = \frac{m_{1}·l}{l_{1}·m_{1}} and p = \frac{m_{1}·l}{m_{1}·l_{1}} and p = \frac{l·m_{1}}{m_{1}·l_{1}} .

(56) Suppose p = \frac{m}{n} and n ≠ 0 and k ≠ 0. Then p = \frac{m·k}{n·k} and p = \frac{k·m}{n·k} and p = \frac{k·m}{k·n} and p = \frac{m·k}{k·n}.

(57) Suppose p = \frac{m}{l} and l ≠ 0 and k ≠ 0. Then p = \frac{m·k}{l·k} and p = \frac{k·m}{l·k} and p = \frac{k·m}{k·l} and p = \frac{m·k}{k·l}.

(58) Suppose p = \frac{l}{n} and n ≠ 0 and k ≠ 0. Then p = \frac{l·k}{n·k} and p = \frac{k·l}{n·k} and p = \frac{k·l}{k·n} and p = \frac{l·k}{k·n}.

(59) Suppose p = \frac{l}{l_{1}} and l_{1} ≠ 0 and k ≠ 0. Then p = \frac{l·k}{l_{1}·k} and p = \frac{k·l}{l_{1}·k} and p = \frac{k·l}{k·l_{1}} and p = \frac{l·k}{k·l_{1}} .

(60) If k ≠ 0 and p = \frac{m}{k}, then there exists l such that m = num p · l and k = den p · l.

(61) If p = \frac{m}{n} and n ≠ 0, then there exists m_{1} such that m = num p · m_{1} and n = den p · m_{1}.

(62) For no l holds 1 < l and there exist m, k such that num p = m · l and den p = k · l.

(63) If p = \frac{m}{k} and k ≠ 0 and for no l holds 1 < l and there exist m_{1}, k_{1} such that m = m_{1} · l and k = k_{1} · l, then k = den p and m = num p.

(64) p < −1 if and only if num p < − den p.

(65) p ≤ −1 if and only if num p ≤ − den p.

(66) p < −1 if and only if den p < − num p.

(67) p ≤ −1 if and only if den p ≤ − num p.

(68) −1 < p if and only if − den p < num p.

(69) p ≥ −1 if and only if num p ≥ − den p.

(70) −1 < p if and only if − num p < den p.

(71) p ≥ −1 if and only if den p ≥ − num p.

(72) p < 1 if and only if num p < den p.

(73) p ≤ 1 if and only if num p ≤ den p.

(74) 1 < p if and only if den p < num p.

(75) p ≥ 1 if and only if num p ≥ den p.

(76) p < 0 if and only if num p < 0.

(77) p ≤ 0 if and only if num p ≤ 0.

(78) 0 < p if and only if 0 < num p.

(79) p ≥ 0 if and only if num p ≥ 0.

(80) a < p if and only if a · den p < num p.

(81) a ≤ p if and only if a · den p ≤ num p.

(82) p < a if and only if num p < a · den p.

(83) a ≥ p if and only if a · den p ≥ num p.

(84) p = q if and only if den p = den q and num p = num q.

(85) If p = \frac{m}{n} and n ≠ 0 and q = \frac{m_{1}}{n_{1}} and n_{1} ≠ 0, then p = q if and only if m · n_{1} = m_{1} · n.

(86) p < q if and only if num p · den q < num q · den p.

(87) den(−p) = den p and num(−p) = − num p.

(88) 0 < p and q = \frac{1}{p} if and only if num q = den p and den q = num p.

(89) p < 0 and q = \frac{1}{p} if and only if num q = − den p and den q = − num p.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[3] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[4] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[5] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

846

Basis of Real Linear Space

Wojciech A. Trybulec Warsaw University

Summary. Notions of linear independence and dependence of set of vectors, the subspace generated by a set of vectors and basis of real linear space are introduced. Some theorems concerning those notion, are proved.

MML Identifier: RLVECT 3.

The papers [6], [2], [1], [3], [11], [4], [10], [9], [5], [8], and [7] provide the notation and terminology for this paper. For simplicity we follow a convention: x is arbitrary, a, b are real numbers, V is a real linear space, W, W_{1}, W_{2}, W_{3} are subspaces of V, v, v_{1}, v_{2} are vectors of V, A, B are subsets of the vectors of V, L, L_{1}, L_{2} are linear combinations of V, l is a linear combination of A, F, G are finite sequences of elements of the vectors of V, f is a function from the vectors of V into , X, Y, Z are sets, M is a non-empty family of sets, and C_{1} is a choice function of M. One can prove the following four propositions:

(1) ∑(L_{1} + L_{2}) = ∑ L_{1} + ∑ L_{2}.

(2) ∑(a · L) = a · ∑ L.

(3) ∑(−L) = − ∑ L.

(4) ∑(L_{1} − L_{2}) = ∑ L_{1} − ∑ L_{2}.

We now define two new predicates. Let us consider V, A. We say that A is linearly independent if and only if: (Def.1) for every l such that ∑ l = 0_{V} holds support l = ∅. We say that A is linearly dependent if and only if A is not linearly independent.

One can prove the following propositions:

(5) A is linearly independent if and only if for every l such that ∑ l = 0_{V} holds support l = ∅.

(6) If A ⊆ B and B is linearly independent, then A is linearly independent.

(7) If A is linearly independent, then 0_{V} ∉ A.

(8) ∅_{the vectors of V} is linearly independent.

(9) {v} is linearly independent if and only if v ≠ 0_{V}.

(10) {0_{V} } is linearly dependent.

(11) If {v_{1}, v_{2}} is linearly independent, then v_{1} ≠ 0_{V} and v_{2} ≠ 0_{V}.

(12) {v, 0_{V} } is linearly dependent and {0_{V} , v} is linearly dependent.

(13) v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent if and only if v_{2} ≠ 0_{V} and for every a holds v_{1} ≠ a · v_{2}.

(14) v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent if and only if for all a, b

such that a · v_{1} + b · v_{2} = 0_{V} holds a = 0 and b = 0. Let us consider V, A. The functor Lin(A) yields a subspace of V and is

defined by: (Def.2) the vectors of Lin(A) = {∑ l}.

We now state four propositions:

(15) If the vectors of W = {∑ l}, then W = Lin(A).

(16) The vectors of Lin(A) = {∑ l}.

(17) x ∈ Lin(A) if and only if there exists l such that x = ∑ l.

(18) If x ∈ A, then x ∈ Lin(A).

The following propositions are true:

(19) Lin(∅_{the vectors of V}) = 0_{V}.

(20) If Lin(A) = 0_{V}, then A = ∅ or A = {0_{V} }.

(21) If A = the vectors of W, then Lin(A) = W.

(22) If A = the vectors of V, then Lin(A) = V.

(23) If A ⊆ B, then Lin(A) is a subspace of Lin(B).

(24) If Lin(A) = V and A ⊆ B, then Lin(B) = V.

(25) Lin(A ∪ B) = Lin(A) + Lin(B).

(26) Lin(A ∩ B) is a subspace of Lin(A) ∩ Lin(B).

(27) If A is linearly independent, then there exists B such that A ⊆ B and B is linearly independent and Lin(B) = V.

(28) If Lin(A) = V, then there exists B such that B ⊆ A and B is linearly

independent and Lin(B) = V. Let us consider V. A subset of the vectors of V is called a basis of V if:

(Def.3) it is linearly independent and Lin(it) = V.

The following proposition is true

(29) If A is linearly independent and Lin(A) = V, then A is a basis of V. In the sequel I is a basis of V. Next we state a number of propositions:

(30) I is linearly independent.

(31) Lin(I) = V.

(32) If A is linearly independent, then there exists I such that A ⊆ I.

(33) If Lin(A) = V, then there exists I such that I ⊆ A.

(34) If Z ≠ ∅ and Z is finite and for all X, Y such that X ∈ Z and Y ∈ Z holds X ⊆ Y or Y ⊆ X, then ⋃ Z ∈ Z.

(35) If ∅ ∉ M, then dom C_{1} = M and rng C_{1} ⊆ ⋃ M.

(36) x ∈ 0_{V} if and only if x = 0_{V}.

(37) If W_{1} is a subspace of W_{3}, then W_{1} ∩ W_{2} is a subspace of W_{3}.

(38) If W_{1} is a subspace of W_{2} and W_{1} is a subspace of W_{3}, then W_{1} is a subspace of W_{2} ∩ W_{3}.

(39) If W_{1} is a subspace of W_{3} and W_{2} is a subspace of W_{3}, then W_{1} + W_{2} is a subspace of W_{3}.

(40) If W_{1} is a subspace of W_{2}, then W_{1} is a subspace of W_{2} + W_{3}.

(41) f · (F ^{}G) = (f · F)^{}(f · G).

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[5] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[6] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[7] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.

[8] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics, 1(2):395–399, 1990.

[9] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

[10] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

850 Wojciech A. Trybulec

[11] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathemat-

ics, 1(2):291–296, 1990. Received July 10, 1990

Finite Sums of Vectors in Vector Space

Wojciech A. Trybulec Warsaw University

Summary. We define the sum of finite sequences of vectors in vector space. Theorems concerning those sums are proved.

MML Identifier: VECTSP 3.

The terminology and notation used here have been introduced in the following papers: [7], [2], [3], [5], [6], [4], and [1]. Let F be a field. An element of F is an element of the carrier of F.

For simplicity we follow a convention: x will be arbitrary, G_{1} will denote a field, a will denote an element of G_{1}, V will denote a vector space over G_{1}, and v, v_{1}, v_{2}, w, u will denote vectors of V. Let us consider G_{1}, V, x. The predicate x ∈ V is defined by: (Def.1) x ∈ the carrier of the carrier of V.

Next we state two propositions:

(1) x ∈ V if and only if x ∈ the carrier of the carrier of V.

(2) v ∈ V.

We follow a convention: F, G, H will be finite sequences of elements of the carrier of the carrier of V, f will be a function from into the carrier of the carrier of V, and i, j, k, n will be natural numbers. Let us consider G_{1}, V, f, j. Then f(j) is a vector of V.

Let us consider G_{1}, V, F. The functor ∑ F yielding a vector of V is defined as follows: (Def.2) there exists f such that ∑ F = f(len F) and f(0) = Θ_{V} and for all j,

v such that j < len F and v = F (j + 1) holds f(j + 1) = f(j) + v.

We now state a number of propositions:

(3) If there exists f such that u = f(len F) and f(0) = Θ_{V} and for all j, v such that j < len F and v = F (j + 1) holds f(j + 1) = f(j) + v, then u = ∑ F.

(4) There exists f such that ∑ F = f(len F) and f(0) = Θ_{V} and for all j, v such that j < len F and v = F (j + 1) holds f(j + 1) = f(j) + v.

(5) If k ∈ Seg n and len F = n, then F (k) is a vector of V.

(6) If len F = len G + 1 and G = F Seg(len G) and v = F (len F), then ∑ F = ∑ G + v.

(7) ∑(F ^{}G) = ∑ F + ∑ G.

(8) If len F = len G and len F = len H and for every k such that k ∈ Seg(len F) holds H(k) = π_{k}F + π_{k}G, then ∑ H = ∑ F + ∑ G.

(9) If len F = len G and for all k, v such that k ∈ Seg(len F) and v = G(k) holds F (k) = a · v, then ∑ F = a · ∑ G.

(10) If len F = len G and for every k such that k ∈ Seg(len F) holds G(k) = a · π_{k}F, then ∑ G = a · ∑ F.

(11) If len F = len G and for all k, v such that k ∈ Seg(len F) and v = G(k) holds F (k) = −v, then ∑ F = − ∑ G.

(12) If len F = len G and for every k such that k ∈ Seg(len F) holds G(k) = −π_{k}F, then ∑ G = − ∑ F.

(13) If len F = len G and len F = len H and for every k such that k ∈ Seg(len F) holds H(k) = π_{k}F − π_{k}G, then ∑ H = ∑ F − ∑ G.

(14) If rng F = rng G and F is one-to-one and G is one-to-one, then ∑ F = ∑ G.

(15) For all F, G and for every permutation f of dom F such that len F = len G and for every i such that i ∈ dom G holds G(i) = F (f(i)) holds ∑ F = ∑ G.

(16) For every permutation f of dom F such that G = F ·f holds ∑ F = ∑ G.

(17) ∑ ε_{the carrier of the carrier of V} = Θ_{V}.

(18) ∑⟨v⟩ = v.

(19) ∑⟨v, u⟩ = v + u.

(20) ∑⟨v, u, w⟩ = (v + u) + w.

(21) a · ∑ ε_{the carrier of the carrier of V} = Θ_{V}.

(22) a · ∑⟨v⟩ = a · v.

(23) a · ∑⟨v, u⟩ = a · v + a · u.

(24) a · ∑⟨v, u, w⟩ = (a · v + a · u) + a · w.

(25) − ∑ ε_{the carrier of the carrier of V} = Θ_{V}.

(26) − ∑⟨v⟩ = −v.

(27) − ∑⟨v, u⟩ = (−v) − u.

(28) − ∑⟨v, u, w⟩ = ((−v) − u) − w.

(29) ∑⟨v, w⟩ = ∑⟨w, v⟩.

(30) ∑⟨v, w⟩ = ∑⟨v⟩ + ∑⟨w⟩.

(31) ∑⟨Θ_{V} , Θ_{V} ⟩ = Θ_{V}.

(32) ∑⟨Θ_{V} , v⟩ = v and ∑⟨v, Θ_{V} ⟩ = v.

(33) ∑⟨v, −v⟩ = Θ_{V} and ∑⟨−v, v⟩ = Θ_{V}.

(34) ∑⟨v, −w⟩ = v − w and ∑⟨−w, v⟩ = v − w.

(35) ∑⟨−v, −w⟩ = −(v + w) and ∑⟨−w, −v⟩ = −(v + w).

(36) ∑⟨u, v, w⟩ = (∑⟨u⟩ + ∑⟨v⟩) + ∑⟨w⟩.

(37) ∑⟨u, v, w⟩ = ∑⟨u, v⟩ + w.

(38) ∑⟨u, v, w⟩ = ∑⟨v, w⟩ + u.

(39) ∑⟨u, v, w⟩ = ∑⟨u, w⟩ + v.

(40) ∑⟨u, v, w⟩ = ∑⟨u, w, v⟩.

(41) ∑⟨u, v, w⟩ = ∑⟨v, u, w⟩.

(42) ∑⟨u, v, w⟩ = ∑⟨v, w, u⟩.

(43) ∑⟨u, v, w⟩ = ∑⟨w, u, v⟩.

(44) ∑⟨u, v, w⟩ = ∑⟨w, v, u⟩.

(45) ∑⟨Θ_{V} , Θ_{V} , Θ_{V} ⟩ = Θ_{V}.

(46) ∑⟨Θ_{V} , Θ_{V} , v⟩ = v and ∑⟨Θ_{V} , v, Θ_{V} ⟩ = v and ∑⟨v, Θ_{V} , Θ_{V} ⟩ = v.

(47) ∑⟨Θ_{V} , u, v⟩ = u + v and ∑⟨u, v, Θ_{V} ⟩ = u + v and ∑⟨u, Θ_{V} , v⟩ = u + v.

(48) If len F = 0, then ∑ F = Θ_{V}.

(49) If len F = 1, then ∑ F = F (1).

(50) If len F = 2 and v_{1} = F (1) and v_{2} = F (2), then ∑ F = v_{1} + v_{2}.

(51) If len F = 3 and v_{1} = F (1) and v_{2} = F (2) and v = F (3), then ∑ F = (v_{1} + v_{2}) + v.

(52) v − v = Θ_{V}.

(53) −(v + w) = (−v) + (−w).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[5] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[6] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

854 Wojciech A. Trybulec

[7] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets.

Formalized Mathematics, 1(1):17–23, 1990. Received July 12, 1990

Subgroup and Cosets of Subgroups

Wojciech A. Trybulec^{1} Warsaw University

Summary. We introduce notion of subgroup, coset of a subgroup, sets of left and right cosets of a subgroup. We define multiplication of two subset of a group, subset of reverse elemens of a group, intersection of two subgroups. We define the notion of an index of a subgroup and prove Lagrange theorem which states that in a finite group the order of the group equals the order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved.

MML Identifier: GROUP 2.

The papers [9], [6], [3], [4], [1], [11], [10], [12], [5], [8], [7], and [2] provide the notation and terminology for this paper. Let D be a non-empty set. Then ∅_{D} is a subset of D. Then Ω_{D} is a subset of D.

For simplicity we adopt the following convention: x is arbitrary, X, Y, Z are sets, k is a natural number, G, G_{1}, G_{2}, G_{3} are groups, and a, b, g, g_{1}, g_{2}, h are elements of G. Let us consider G. A subset of G is a subset of the carrier of G.

In the sequel A, B, C denote subsets of G. The following propositions are true:

(1) If x ∈ A, then x ∈ G.

(2) If x ∈ A, then x is an element of G.

(3) If G is finite, then A is finite.

Let us consider G, A. The functor A^{−1} yielding a subset of G is defined by: (Def.1) A^{−1} = {g^{−1} : g ∈ A}.

Next we state several propositions:

(4) A^{−1} = {g^{−1} : g ∈ A}.

(5) x ∈ A^{−1} if and only if there exists g such that x = g^{−1} and g ∈ A.

(6) {g}^{−1} = {g^{−1}}.

1Supported by RPBP.III-24.C1

(7) {g, h}^{−1} = {g^{−1}, h^{−1}}.

(8) (∅_{the carrier of G})^{−1} = ∅.

(9) (Ω_{the carrier of G})^{−1} = the carrier of G.

(10) A ≠ ∅ if and only if A^{−1} ≠ ∅.

Let us consider G, A, B. The functor A · B yielding a subset of G is defined as follows: (Def.2) A · B = {g · h : g ∈ A ∧ h ∈ B}.

One can prove the following propositions:

(11) A · B = {g · h : g ∈ A ∧ h ∈ B}.

(12) x ∈ A · B if and only if there exist g, h such that x = g · h and g ∈ A and h ∈ B.

(13) A ≠ ∅ and B ≠ ∅ if and only if A · B ≠ ∅.

(14) (A · B) · C = A · (B · C).

(15) (A · B)^{−1} = B^{−1} · A^{−1}.

(16) A · (B ∪ C) = A · B ∪ A · C.

(17) (A ∪ B) · C = A · C ∪ B · C.

(18) A · (B ∩ C) ⊆ (A · B) ∩ (A · C).

(19) (A ∩ B) · C ⊆ (A · C) ∩ (B · C).

(20) ∅_{the carrier of G} · A = ∅ and A · ∅_{the carrier of G} = ∅.

(21) If A ≠ ∅, then Ω_{the carrier of G} · A = the carrier of G and A · Ω_{the carrier of G} = the carrier of G.

(22) {g} · {h} = {g · h}.

(23) {g} · {g_{1}, g_{2}} = {g · g_{1}, g · g_{2}}.

(24) {g_{1}, g_{2}} · {g} = {g_{1} · g, g_{2} · g}.

(25) {g, h} · {g_{1}, g_{2}} = {g · g_{1}, g · g_{2}, h · g_{1}, h · g_{2}}.

(26) If for all g_{1}, g_{2} such that g_{1} ∈ A and g_{2} ∈ A holds g_{1} · g_{2} ∈ A and for every g such that g ∈ A holds g^{−1} ∈ A, then A · A = A.

(27) If for every g such that g ∈ A holds g^{−1} ∈ A, then A^{−1} = A.

(28) If for all a, b such that a ∈ A and b ∈ B holds a · b = b · a, then A · B = B · A.

(29) If G is an Abelian group, then A · B = B · A.

(30) If G is an Abelian group, then (A · B)^{−1} = A^{−1} · B^{−1}.

We now define two new functors. Let us consider G, g, A. The functor g · A yields a subset of G and is defined as follows: (Def.3) g · A = {g} · A. The functor A · g yielding a subset of G is defined as follows: (Def.4) A · g = A · {g}.

Next we state a number of propositions:

(31) g · A = {g} · A.

(32) A · g = A · {g}.

(33) x ∈ g · A if and only if there exists h such that x = g · h and h ∈ A.

(34) x ∈ A · g if and only if there exists h such that x = h · g and h ∈ A.

(35) (g · A) · B = g · (A · B).

(36) (A · g) · B = A · (g · B).

(37) (A · B) · g = A · (B · g).

(38) (g · h) · A = g · (h · A).

(39) (g · A) · h = g · (A · h).

(40) (A · g) · h = A · (g · h).

(41) ∅_{the carrier of G} · a = ∅ and a · ∅_{the carrier of G} = ∅.

(42) Ω_{the carrier of G} · a = the carrier of G and a · Ω_{the carrier of G} = the carrier of G.

(43) (1_{G}) · A = A and A · (1_{G}) = A.

(44) If G is an Abelian group, then g · A = A · g.

Let us consider G. A group is said to be a subgroup of G if: (Def.5) the carrier of it ⊆ the carrier of G and the operation of it = (the

operation of G) [: the carrier of it, the carrier of it :]. One can prove the following proposition

(45) If the carrier of G_{1} ⊆ the carrier of G_{2} and the operation of G_{1} = (the operation of G_{2}) [: the carrier of G_{1}, the carrier of G_{1} :], then G_{1} is a subgroup of G_{2}.

We follow the rules: I, H, H_{1}, H_{2}, H_{3} will be subgroups of G and h, h_{1}, h_{2} will be elements of H. One can prove the following propositions:

(46) The carrier of H ⊆ the carrier of G.

(47) The operation of H = (the operation of G) [: the carrier of H, the carrier of H:].

(48) If G is finite, then H is finite.

(49) If x ∈ H, then x ∈ G.

(50) h ∈ G.

(51) h is an element of G.

(52) If h_{1} = g_{1} and h_{2} = g_{2}, then h_{1} · h_{2} = g_{1} · g_{2}.

(53) 1_{H} = 1_{G}.

(54) 1_{H_{1}} = 1_{H_{2}}.

(55) 1_{G} ∈ H.

(56) 1_{H_{1}} ∈ H_{2}.

(57) If h = g, then h^{−1} = g^{−1}.

(58) ·_{H}^{−1} = ·_{G}^{−1} (the carrier of H).

(59) If g_{1} ∈ H and g_{2} ∈ H, then g_{1} · g_{2} ∈ H.

(60) If g ∈ H, then g^{−1} ∈ H.

(61) If A ≠ ∅ and for all g_{1}, g_{2} such that g_{1} ∈ A and g_{2} ∈ A holds g_{1} · g_{2} ∈ A and for every g such that g ∈ A holds g^{−1} ∈ A, then there exists H such that the carrier of H = A.

(62) If G is an Abelian group, then H is an Abelian group.

Let G be an Abelian group. We see that the subgroup of G is an Abelian group.

We now state several propositions:

(63) G is a subgroup of G.

(64) If G_{1} is a subgroup of G_{2} and G_{2} is a subgroup of G_{1}, then G_{1} = G_{2}.

(65) If G_{1} is a subgroup of G_{2} and G_{2} is a subgroup of G_{3}, then G_{1} is a subgroup of G_{3}.

(66) If the carrier of H_{1} ⊆ the carrier of H_{2}, then H_{1} is a subgroup of H_{2}.

(67) If for every g such that g ∈ H_{1} holds g ∈ H_{2}, then H_{1} is a subgroup of H_{2}.

(68) If the carrier of H_{1} = the carrier of H_{2}, then H_{1} = H_{2}.

(69) If for every g holds g ∈ H_{1} if and only if g ∈ H_{2}, then H_{1} = H_{2}. Let us consider G, H_{1}, H_{2}. Let us note that one can characterize the predicate

H_{1} = H_{2} by the following (equivalent) condition: (Def.6) for every g holds g ∈ H_{1} if and only if g ∈ H_{2}.

The following two propositions are true:

(70) If the carrier of H = the carrier of G, then H = G.

(71) If for every g holds g ∈ H, then H = G.

Let us consider G. The functor {1}_{G} yields a subgroup of G and is defined by: (Def.7) the carrier of {1}_{G} = {1_{G}}.

Let us consider G. The functor Ω_{G} yielding a subgroup of G is defined as follows: (Def.8) Ω_{G} = G.

The following propositions are true:

(72) If the carrier of H = {1_{G}}, then H = {1}_{G}.

(73) The carrier of {1}_{G} = {1_{G}}.

(74) Ω_{G} = G.

(75) {1}_{H} = {1}_{G}.

(76) {1}_{H_{1}} = {1}_{H_{2}}.

(77) {1}_{G} is a subgroup of H.

(78) H is a subgroup of Ω_{G}.

(79) G is a subgroup of Ω_{G}.

(80) {1}_{G} is finite.

(81) ord({1}_{G}) = 1.

(82) If H is finite and ord(H) = 1, then H = {1}_{G}.

(83) Ord(H) ≤ Ord(G).

(84) If G is finite, then ord(H) ≤ ord(G).

(85) If G is finite and ord(G) = ord(H), then H = G.

Let us consider G, H. The functor \overline{H} yields a subset of G and is defined by: (Def.9) \overline{H} = the carrier of H.

The following propositions are true:

(86) \overline{H} = the carrier of H.

(87) \overline{H} ≠ ∅.

(88) x ∈ \overline{H} if and only if x ∈ H.

(89) If g_{1} ∈ \overline{H} and g_{2} ∈ \overline{H}, then g_{1} · g_{2} ∈ \overline{H}.

(90) If g ∈ \overline{H}, then g^{−1} ∈ \overline{H}.

(91) \overline{H} · \overline{H} = \overline{H}.

(92) \overline{H}^{−1} = \overline{H}.

(93) \overline{H_{1}} · \overline{H_{2}} = \overline{H_{2}} · \overline{H_{1}} if and only if there exists H such that the carrier of H = \overline{H_{1}} · \overline{H_{2}}.

(94) If G is an Abelian group, then there exists H such that the carrier of

H = \overline{H_{1}} · \overline{H_{2}}. Let us consider G, H_{1}, H_{2}. The functor H_{1} ∩ H_{2} yields a subgroup of G and

is defined as follows: (Def.10) the carrier of H_{1} ∩ H_{2} = \overline{H_{1}} ∩ \overline{H_{2}}.

One can prove the following propositions:

(95) If the carrier of H = \overline{H_{1}} ∩ \overline{H_{2}}, then H = H_{1} ∩ H_{2}.

(96) The carrier of H_{1} ∩ H_{2} = \overline{H_{1}} ∩ \overline{H_{2}}.

(97) H = H_{1} ∩ H_{2} if and only if the carrier of H = (the carrier of H_{1})∩ (the carrier of H_{2}).

(98) \overline{H_{1} ∩ H_{2}} = \overline{H_{1}} ∩ \overline{H_{2}}.

(99) x ∈ H_{1} ∩ H_{2} if and only if x ∈ H_{1} and x ∈ H_{2}.

(100) H ∩ H = H.

(101) H_{1} ∩ H_{2} = H_{2} ∩ H_{1}.

(102) (H_{1} ∩ H_{2}) ∩ H_{3} = H_{1} ∩ (H_{2} ∩ H_{3}).

(103) {1}_{G} ∩ H = {1}_{G} and H ∩ {1}_{G} = {1}_{G}.

(104) H ∩ Ω_{G} = H and Ω_{G} ∩ H = H.

(105) Ω_{G} ∩ Ω_{G} = G.

(106) H_{1} ∩ H_{2} is a subgroup of H_{1} and H_{1} ∩ H_{2} is a subgroup of H_{2}.

(107) H_{1} is a subgroup of H_{2} if and only if H_{1} ∩ H_{2} = H_{1}.

(108) If H_{1} is a subgroup of H_{2}, then H_{1} ∩ H_{3} is a subgroup of H_{2}.

(109) If H_{1} is a subgroup of H_{2} and H_{1} is a subgroup of H_{3}, then H_{1} is a subgroup of H_{2} ∩ H_{3}.

(110) If H_{1} is a subgroup of H_{2}, then H_{1} ∩ H_{3} is a subgroup of H_{2} ∩ H_{3}.

(111) If H_{1} is finite or H_{2} is finite, then H_{1} ∩ H_{2} is finite.

We now define two new functors. Let us consider G, H, A. The functor A·H yielding a subset of G is defined as follows:

(Def.11) A · H = A · \overline{H}.

The functor H · A yields a subset of G and is defined as follows: (Def.12) H · A = \overline{H} · A.

One can prove the following propositions:

(112) A · H = A · \overline{H}.

(113) H · A = \overline{H} · A.

(114) x ∈ A · H if and only if there exist g_{1}, g_{2} such that x = g_{1} · g_{2} and g_{1} ∈ A and g_{2} ∈ H.

(115) x ∈ H · A if and only if there exist g_{1}, g_{2} such that x = g_{1} · g_{2} and g_{1} ∈ H and g_{2} ∈ A.

(116) (A · B) · H = A · (B · H).

(117) (A · H) · B = A · (H · B).

(118) (H · A) · B = H · (A · B).

(119) (A · H_{1}) · H_{2} = A · (H_{1} · \overline{H_{2}}).

(120) (H_{1} · A) · H_{2} = H_{1} · (A · H_{2}).

(121) (H_{1} · \overline{H_{2}}) · A = H_{1} · (H_{2} · A).

(122) If G is an Abelian group, then A · H = H · A.

We now define two new functors. Let us consider G, H, a. The functor a · H yielding a subset of G is defined as follows:

(Def.13) a · H = a · \overline{H}.

The functor H · a yielding a subset of G is defined by: (Def.14) H · a = \overline{H} · a.

The following propositions are true:

(123) a · H = a · \overline{H}.

(124) H · a = \overline{H} · a.

(125) x ∈ a · H if and only if there exists g such that x = a · g and g ∈ H.

(126) x ∈ H · a if and only if there exists g such that x = g · a and g ∈ H.

(127) (a · b) · H = a · (b · H).

(128) (a · H) · b = a · (H · b).

(129) (H · a) · b = H · (a · b).

(130) a ∈ a · H and a ∈ H · a.

(131) a · H ≠ ∅ and H · a ≠ ∅.

(132) (1_{G}) · H = \overline{H} and H · (1_{G}) = \overline{H}.

(133) {1}_{G} · a = {a} and a · {1}_{G} = {a}.

(134) a · Ω_{G} = the carrier of G and Ω_{G} · a = the carrier of G.

(135) If G is an Abelian group, then a · H = H · a.

(136) a ∈ H if and only if a · H = \overline{H}.

(137) a · H = b · H if and only if b^{−1} · a ∈ H.

(138) a · H = b · H if and only if a · H meets b · H.

(139) (a · b) · H ⊆ (a · H) · (b · H).

(140) \overline{H} ⊆ (a · H) · (a^{−1} · H) and \overline{H} ⊆ (a^{−1} · H) · (a · H).

(141) a^{2} · H ⊆ (a · H) · (a · H).

(142) a ∈ H if and only if H · a = \overline{H}.

(143) H · a = H · b if and only if b · a^{−1} ∈ H.

(144) H · a = H · b if and only if H · a meets H · b.

(145) (H · a) · b ⊆ (H · a) · (H · b).

(146) \overline{H} ⊆ (H · a) · (H · a^{−1}) and \overline{H} ⊆ (H · a^{−1}) · (H · a).

(147) H · a^{2} ⊆ (H · a) · (H · a).

(148) a · (H_{1} ∩ H_{2}) = (a · H_{1}) ∩ (a · H_{2}).

(149) (H_{1} ∩ H_{2}) · a = (H_{1} · a) ∩ (H_{2} · a).

(150) There exists H_{1} such that the carrier of H_{1} = (a · H_{2}) · a^{−1}.

(151) a · H ≈ b · H.

(152) a · H ≈ H · b.

(153) H · a ≈ H · b.

(154) \overline{H} ≈ a · H and \overline{H} ≈ H · a.

(155) Ord(H) = \overline{\overline{a · H}} and Ord(H) = \overline{\overline{H · a}}.

(156) If H is finite, then ord(H) = card(a · H) and ord(H) = card(H · a). The scheme SubFamComp deals with a set A, a family B of subsets of A, a

family C of subsets of A, and a unary predicate P, and states that:

B = C provided the parameters meet the following requirements:

• for every subset X of A holds X ∈ B if and only if P[X],

• for every subset X of A holds X ∈ C if and only if P[X].

We now define two new functors. Let us consider G, H. The left cosets of H yielding a family of subsets of the carrier of G is defined as follows:

(Def.15) A ∈ the left cosets of H if and only if there exists a such that A = a · H. The right cosets of H yielding a family of subsets of the carrier of G is defined by:

(Def.16) A ∈ the right cosets of H if and only if there exists a such that A = H ·a.

In the sequel F denotes a family of subsets of the carrier of G. One can prove the following propositions:

(157) If for every A holds A ∈ F if and only if there exists a such that A = a · H, then F = the left cosets of H.

(158) If for every A holds A ∈ F if and only if there exists a such that A = H · a, then F = the right cosets of H.

(159) A ∈ the left cosets of H if and only if there exists a such that A = a · H.

(160) A ∈ the right cosets of H if and only if there exists a such that A = H ·a.

(161) If x ∈ the left cosets of H or x ∈ the right cosets of H, then x is a subset of G.

(162) x ∈ the left cosets of H if and only if there exists a such that x = a · H.

(163) x ∈ the right cosets of H if and only if there exists a such that x = H ·a.

(164) If G is finite, then the right cosets of H is finite and the left cosets of H is finite.

(165) \overline{H} ∈ the left cosets of H and \overline{H} ∈ the right cosets of H.

(166) The left cosets of H ≈ the right cosets of H.

(167) ⋃(The left cosets of H) = the carrier of G and ⋃(the right cosets of H) = the carrier of G.

(168) The left cosets of{1}_{G} = {{a}}.

(169) The right cosets of{1}_{G} = {{a}}.

(170) If the left cosets of H = {{a}}, then H = {1}_{G}.

(171) If the right cosets of H = {{a}}, then H = {1}_{G}.

(172) The left cosets of Ω_{G} = { the carrier of G} and the right cosets of Ω_{G} = { the carrier of G}.

(173) If the left cosets of H = { the carrier of G}, then H = G.

(174) If the right cosets of H = { the carrier of G}, then H = G.

Let us consider G, H. The functor |• : H| yielding a cardinal number is defined by:

(Def.17) |• : H| = \overline{\overline{the left cosets of H}}.

We now state the proposition

(175) |• : H| = \overline{\overline{the left cosets of H}} and |• : H| = \overline{\overline{the right cosets of H}}. Let us consider G, H. Let us assume that the left cosets of H is finite. The

functor |• : H|_{}yielding a natural number is defined as follows: (Def.18) |• : H|_{}= card(the left cosets of H).

Next we state the proposition

(176) If the left cosets of H is finite, then |• : H|_{}= card(the left cosets of H)

and |• : H|_{}= card(the right cosets of H). Let D be a non-empty set, and let d be an element of D. Then {d} is an

element of Fin D.

The following two propositions are true:

(177) If G is finite, then ord(G) = ord(H) · |• : H|_{}.

(178) If G is finite, then ord(H) | ord(G).

In the sequel J will denote a subgroup of H. One can prove the following propositions:

(179) If G is finite and I = J, then |• : I|_{}= |• : J|_{}· |• : H|_{}.

(180) |• : Ω_{G}|_{}= 1.

(181) If the left cosets of H is finite and |• : H|_{}= 1, then H = G.

(182) |• : {1}_{G}| = Ord(G).

(183) If G is finite, then |• : {1}_{G}|_{}= ord(G).

(184) If G is finite and |• : H|_{}= ord(G), then H = {1}_{G}.

(185) If the left cosets of H is finite and |• : H| = Ord(G), then G is finite and H = {1}_{G}.

(186) If X is finite and for every Y such that Y ∈ X holds Y is finite and card Y = k and for every Z such that Z ∈ X and Y ≠ Z holds Y ≈ Z and Y misses Z, then card(⋃ X) = k · card X.

(187) If Y is finite and X ⊆ Y and card X = card Y, then X = Y.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[6] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[7] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[8] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[9] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[10] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

[11] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 864 Wojciech A. Trybulec

[12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics,

1(1):67–71, 1990. Received July 23, 1990

Subspaces and Cosets of Subspaces in Vector Space

Wojciech A. Trybulec^{1} Warsaw University

Summary. We introduce the notions of subspace of vector space and coset of a subspace. We prove a number of theorems concerning those notions. Some theorems that belong rather to [1] are proved.

MML Identifier: VECTSP 4.

The articles [3], [5], [2], [1], and [4] provide the terminology and notation for this paper. For simplicity we adopt the following rules: G_{1} will denote a field, V, X, Y will denote vector spaces over G_{1}, u, v, v_{1}, v_{2} will denote vectors of V, a, b, c will denote elements of G_{1}, and x will be arbitrary. Let us consider G_{1}, V. A subset of V is a subset of the carrier of the carrier of V.

In the sequel V_{1}, V_{2}, V_{3} denote subsets of V. Let us consider G_{1}, V, V_{1}. We say that V_{1} is linearly closed if and only if: (Def.1) for all v, u such that v ∈ V_{1} and u ∈ V_{1} holds v + u ∈ V_{1} and for all a,

v such that v ∈ V_{1} holds a · v ∈ V_{1}. The following propositions are true:

(1) If for all v, u such that v ∈ V_{1} and u ∈ V_{1} holds v + u ∈ V_{1} and for all a, v such that v ∈ V_{1} holds a · v ∈ V_{1}, then V_{1} is linearly closed.

(2) If V_{1} is linearly closed, then for all v, u such that v ∈ V_{1} and u ∈ V_{1} holds v + u ∈ V_{1}.

(3) If V_{1} is linearly closed, then for all a, v such that v ∈ V_{1} holds a·v ∈ V_{1}.

(4) If V_{1} ≠ ∅ and V_{1} is linearly closed, then Θ_{V} ∈ V_{1}.

(5) If V_{1} is linearly closed, then for every v such that v ∈ V_{1} holds −v ∈ V_{1}.

(6) If V_{1} is linearly closed, then for all v, u such that v ∈ V_{1} and u ∈ V_{1} holds v − u ∈ V_{1}.

(7) {Θ_{V} } is linearly closed.

1Supported by RPBP.III-24.C1

(8) If the carrier of the carrier of V = V_{1}, then V_{1} is linearly closed.

(9) If V_{1} is linearly closed and V_{2} is linearly closed and V_{3} = {v + u : v ∈ V_{1} ∧ u ∈ V_{2}}, then V_{3} is linearly closed.

(10) If V_{1} is linearly closed and V_{2} is linearly closed, then V_{1} ∩ V_{2} is linearly

closed. Let us consider G_{1}, V. A vector space over G_{1} is said to be a subspace of V

if: (Def.2) the carrier of the carrier of it ⊆ the carrier of the carrier of V and the

zero of the carrier of it = the zero of the carrier of V and the addition of the carrier of it = (the addition of the carrier of V) [: the carrier of the carrier of it, the carrier of the carrier of it :] and the multiplication of it = (the multiplication of V) [: the carrier of G_{1}, the carrier of the carrier of it :].

Next we state the proposition

(11) If the carrier of the carrier of X ⊆ the carrier of the carrier of V and the zero of the carrier of X = the zero of the carrier of V and the addition of the carrier of X = (the addition of the carrier of V) [: the carrier of the carrier of X, the carrier of the carrier of X:] and the multiplication of X = (the multiplication of V) [: the carrier of G_{1}, the carrier of the carrier of X:], then X is a subspace of V.

We adopt the following convention: W, W_{1}, W_{2} will be subspaces of V and w, w_{1}, w_{2} will be vectors of W. Next we state a number of propositions:

(12) The carrier of the carrier of W ⊆ the carrier of the carrier of V.

(13) The zero of the carrier of W = the zero of the carrier of V.

(14) The addition of the carrier of W = (the addition of the carrier of V) [: the carrier of the carrier of W, the carrier of the carrier of W :].

(15) The multiplication of W = (the multiplication of V) [: the carrier of G_{1}, the carrier of the carrier of W :].

(16) If x ∈ W_{1} and W_{1} is a subspace of W_{2}, then x ∈ W_{2}.

(17) If x ∈ W, then x ∈ V.

(18) w is a vector of V.

(19) Θ_{W} = Θ_{V}.

(20) Θ_{W_{1}} = Θ_{W_{2}}.

(21) If w_{1} = v and w_{2} = u, then w_{1} + w_{2} = v + u.

(22) If w = v, then a · w = a · v.

(23) If w = v, then −v = −w.

(24) If w_{1} = v and w_{2} = u, then w_{1} − w_{2} = v − u.

(25) Θ_{V} ∈ W.

(26) Θ_{W_{1}} ∈ W_{2}.

(27) Θ_{W} ∈ V.

(28) If u ∈ W and v ∈ W, then u + v ∈ W.

(29) If v ∈ W, then a · v ∈ W.

(30) If v ∈ W, then −v ∈ W.

(31) If u ∈ W and v ∈ W, then u − v ∈ W.

(32) V is a subspace of V.

(33) If V is a subspace of X and X is a subspace of V, then V = X.

(34) If V is a subspace of X and X is a subspace of Y, then V is a subspace of Y.

(35) If the carrier of the carrier of W_{1} ⊆ the carrier of the carrier of W_{2}, then W_{1} is a subspace of W_{2}.

(36) If for every v such that v ∈ W_{1} holds v ∈ W_{2}, then W_{1} is a subspace of W_{2}.

(37) If the carrier of the carrier of W_{1} = the carrier of the carrier of W_{2}, then W_{1} = W_{2}.

(38) If for every v holds v ∈ W_{1} if and only if v ∈ W_{2}, then W_{1} = W_{2}.

(39) If the carrier of the carrier of W = the carrier of the carrier of V, then W = V.

(40) If for every v holds v ∈ W, then W = V.

(41) If the carrier of the carrier of W = V_{1}, then V_{1} is linearly closed.

(42) If V_{1} ≠ ∅ and V_{1} is linearly closed, then there exists W such that V_{1} =

the carrier of the carrier of W. Let us consider G_{1}, V. The functor 0_{V} yielding a subspace of V is defined

by: (Def.3) the carrier of the carrier of 0_{V} = {Θ_{V} }.

Let us consider G_{1}, V. The functor Ω_{V} yields a subspace of V and is defined by: (Def.4) Ω_{V} = V.

The following propositions are true:

(43) The carrier of the carrier of 0_{V} = {Θ_{V} }.

(44) If the carrier of the carrier of W = {Θ_{V} }, then W = 0_{V}.

(45) Ω_{V} = V.

(46) x ∈ 0_{V} if and only if x = Θ_{V}.

(47) 0_{W} = 0_{V}.

(48) 0_{W_{1}} = 0_{W_{2}}.

(49) 0_{W} is a subspace of V.

(50) 0_{V} is a subspace of W.

(51) 0_{W_{1}} is a subspace of W_{2}.

(52) W is a subspace of Ω_{V}.

(53) V is a subspace of Ω_{V}.

Let us consider G_{1}, V, v, W. The functor v + W yielding a subset of V is defined by:

(Def.5) v + W = {v + u : u ∈ W }.

Let us consider G_{1}, V, W. A subset of V is said to be a coset of W if: (Def.6) there exists v such that it = v + W.

In the sequel B, C will denote cosets of W. The following propositions are true:

(54) v + W = {v + u : u ∈ W }.

(55) There exists v such that C = v + W.

(56) If V_{1} = v + W, then V_{1} is a coset of W.

(57) x ∈ v + W if and only if there exists u such that u ∈ W and x = v + u.

(58) Θ_{V} ∈ v + W if and only if v ∈ W.

(59) v ∈ v + W.

(60) Θ_{V} + W = the carrier of the carrier of W.

(61) v + 0_{V} = {v}.

(62) v + Ω_{V} = the carrier of the carrier of V.

(63) Θ_{V} ∈ v + W if and only if v + W = the carrier of the carrier of W.

(64) v ∈ W if and only if v + W = the carrier of the carrier of W.

(65) If v ∈ W, then a · v + W = the carrier of the carrier of W.

(66) If a ≠ 0_{G_{1}} and a · v + W = the carrier of the carrier of W, then v ∈ W.

(67) v ∈ W if and only if (−v) + W = the carrier of the carrier of W.

(68) u ∈ W if and only if v + W = (v + u) + W.

(69) u ∈ W if and only if v + W = (v − u) + W.

(70) v ∈ u + W if and only if u + W = v + W.

(71) If u ∈ v_{1} + W and u ∈ v_{2} + W, then v_{1} + W = v_{2} + W.

(72) If a ≠ 1_{G_{1}} and a · v ∈ v + W, then v ∈ W.

(73) If v ∈ W, then a · v ∈ v + W.

(74) If v ∈ W, then −v ∈ v + W.

(75) u + v ∈ v + W if and only if u ∈ W.

(76) v − u ∈ v + W if and only if u ∈ W.

(77) u ∈ v+W if and only if there exists v_{1} such that v_{1} ∈ W and u = v+v_{1}.

(78) u ∈ v+W if and only if there exists v_{1} such that v_{1} ∈ W and u = v−v_{1}.

(79) There exists v such that v_{1} ∈ v + W and v_{2} ∈ v + W if and only if v_{1} − v_{2} ∈ W.

(80) If v +W = u+W, then there exists v_{1} such that v_{1} ∈ W and v +v_{1} = u.

(81) If v +W = u+W, then there exists v_{1} such that v_{1} ∈ W and v −v_{1} = u.

(82) v + W_{1} = v + W_{2} if and only if W_{1} = W_{2}.

(83) If v + W_{1} = u + W_{2}, then W_{1} = W_{2}.

In the sequel C_{1} denotes a coset of W_{1} and C_{2} denotes a coset of W_{2}. One can prove the following propositions:

(84) There exists C such that v ∈ C.

(85) C is linearly closed if and only if C = the carrier of the carrier of W.

(86) If C_{1} = C_{2}, then W_{1} = W_{2}.

(87) {v} is a coset of 0_{V}.

(88) If V_{1} is a coset of 0_{V}, then there exists v such that V_{1} = {v}.

(89) The carrier of the carrier of W is a coset of W.

(90) The carrier of the carrier of V is a coset of Ω_{V}.

(91) If V_{1} is a coset of Ω_{V}, then V_{1} = the carrier of the carrier of V.

(92) Θ_{V} ∈ C if and only if C = the carrier of the carrier of W.

(93) u ∈ C if and only if C = u + W.

(94) If u ∈ C and v ∈ C, then there exists v_{1} such that v_{1} ∈ W and u+v_{1} = v.

(95) If u ∈ C and v ∈ C, then there exists v_{1} such that v_{1} ∈ W and u−v_{1} = v.

(96) There exists C such that v_{1} ∈ C and v_{2} ∈ C if and only if v_{1} − v_{2} ∈ W.

(97) If u ∈ B and u ∈ C, then B = C.

In the sequel w will denote a vector of V. One can prove the following propositions: (99)^{2} (u + v) − w = u + (v − w).

(100) −(−v) = v.

(101) v − (u − w) = (v − u) + w.

(102) If v + u = v or u + v = v, then u = Θ_{V}.

(103) (a − b) · v = a · v − b · v.

(104) a − 0_{G_{1}} = a.

(105) a − a = 0_{G_{1}}.

(106) a − (b − c) = (a − b) + c.

References

[1] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[2] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[3] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[4] Wojciech A. Trybulec. Finite sums of vectors in vector space. Formalized Mathematics, 1(5):851–854, 1990.

2The proposition (98) became obvious.

870 Wojciech A. Trybulec

[5] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets.

Formalized Mathematics, 1(1):17–23, 1990. Received July 27, 1990

Operations on Subspaces in Vector Space

Wojciech A. Trybulec^{1} Warsaw University

Summary. Sum, direct sum and intersection of subspaces are introduced. We prove some theorems concerning those notions and the decomposition of vector onto two subspaces. Linear complement of a subspace is also defined. We prove theorem that belong rather to [3].

MML Identifier: VECTSP 5.

The papers [2], [8], [9], [5], [3], [4], [6], [1], and [7] provide the terminology and notation for this paper. For simplicity we adopt the following rules: G_{1} will denote a field, V will denote a vector space over G_{1}, W, W_{1}, W_{2}, W_{3} will denote subspaces of V, u, u_{1}, u_{2}, v, v_{1}, v_{2} will denote vectors of V, and x will be arbitrary. Let us consider G_{1}, V, W_{1}, W_{2}. The functor W_{1} + W_{2} yields a subspace of V and is defined by: (Def.1) the carrier of the carrier of W_{1} + W_{2} = {v + u : v ∈ W_{1} ∧ u ∈ W_{2}}.

Let us consider G_{1}, V, W_{1}, W_{2}. The functor W_{1} ∩ W_{2} yields a subspace of V and is defined by: (Def.2) the carrier of the carrier of W_{1} ∩ W_{2} = (the carrier of the carrier of

W_{1})∩ (the carrier of the carrier of W_{2}). We now state a number of propositions:

(1) The carrier of the carrier of W_{1} + W_{2} = {v + u : v ∈ W_{1} ∧ u ∈ W_{2}}.

(2) If the carrier of the carrier of W = {v + u : v ∈ W_{1} ∧ u ∈ W_{2}}, then W = W_{1} + W_{2}.

(3) The carrier of the carrier of W_{1} ∩ W_{2} = (the carrier of the carrier of W_{1})∩ (the carrier of the carrier of W_{2}).

(4) If the carrier of the carrier of W = (the carrier of the carrier of W_{1})∩ (the carrier of the carrier of W_{2}), then W = W_{1} ∩ W_{2}.

(5) x ∈ W_{1} + W_{2} if and only if there exist v_{1}, v_{2} such that v_{1} ∈ W_{1} and v_{2} ∈ W_{2} and x = v_{1} + v_{2}.

1Supported by RPBP.III-24.C1

(6) If v ∈ W_{1} or v ∈ W_{2}, then v ∈ W_{1} + W_{2}.

(7) x ∈ W_{1} ∩ W_{2} if and only if x ∈ W_{1} and x ∈ W_{2}.

(8) W + W = W.

(9) W_{1} + W_{2} = W_{2} + W_{1}.

(10) W_{1} + (W_{2} + W_{3}) = (W_{1} + W_{2}) + W_{3}.

(11) W_{1} is a subspace of W_{1} + W_{2} and W_{2} is a subspace of W_{1} + W_{2}.

(12) W_{1} is a subspace of W_{2} if and only if W_{1} + W_{2} = W_{2}.

(13) 0_{V} + W = W and W + 0_{V} = W.

(14) 0_{V} + Ω_{V} = V and Ω_{V} + 0_{V} = V.

(15) Ω_{V} + W = V and W + Ω_{V} = V.

(16) Ω_{V} + Ω_{V} = V.

(17) W ∩ W = W.

(18) W_{1} ∩ W_{2} = W_{2} ∩ W_{1}.

(19) W_{1} ∩ (W_{2} ∩ W_{3}) = (W_{1} ∩ W_{2}) ∩ W_{3}.

(20) W_{1} ∩ W_{2} is a subspace of W_{1} and W_{1} ∩ W_{2} is a subspace of W_{2}.

(21) W_{1} is a subspace of W_{2} if and only if W_{1} ∩ W_{2} = W_{1}.

(22) If W_{1} is a subspace of W_{2}, then W_{1} ∩ W_{3} is a subspace of W_{2} ∩ W_{3}.

(23) If W_{1} is a subspace of W_{3}, then W_{1} ∩ W_{2} is a subspace of W_{3}.

(24) If W_{1} is a subspace of W_{2} and W_{1} is a subspace of W_{3}, then W_{1} is a subspace of W_{2} ∩ W_{3}.

(25) 0_{V} ∩ W = 0_{V} and W ∩ 0_{V} = 0_{V}.

(26) 0_{V} ∩ Ω_{V} = 0_{V} and Ω_{V} ∩ 0_{V} = 0_{V}.

(27) Ω_{V} ∩ W = W and W ∩ Ω_{V} = W.

(28) Ω_{V} ∩ Ω_{V} = V.

(29) W_{1} ∩ W_{2} is a subspace of W_{1} + W_{2}.

(30) W_{1} ∩ W_{2} + W_{2} = W_{2}.

(31) W_{1} ∩ (W_{1} + W_{2}) = W_{1}.

(32) W_{1} ∩ W_{2} + W_{2} ∩ W_{3} is a subspace of W_{2} ∩ (W_{1} + W_{3}).

(33) If W_{1} is a subspace of W_{2}, then W_{2} ∩ (W_{1} + W_{3}) = W_{1} ∩ W_{2} + W_{2} ∩ W_{3}.

(34) W_{2} + W_{1} ∩ W_{3} is a subspace of (W_{1} + W_{2}) ∩ (W_{2} + W_{3}).

(35) If W_{1} is a subspace of W_{2}, then W_{2}+W_{1}∩W_{3} = (W_{1}+W_{2})∩(W_{2}+W_{3}).

(36) If W_{1} is a subspace of W_{3}, then W_{1} + W_{2} ∩ W_{3} = (W_{1} + W_{2}) ∩ W_{3}.

(37) W_{1} + W_{2} = W_{2} if and only if W_{1} ∩ W_{2} = W_{1}.

(38) If W_{1} is a subspace of W_{2}, then W_{1} + W_{3} is a subspace of W_{2} + W_{3}.

(39) If W_{1} is a subspace of W_{2}, then W_{1} is a subspace of W_{2} + W_{3}.

(40) If W_{1} is a subspace of W_{3} and W_{2} is a subspace of W_{3}, then W_{1} + W_{2} is a subspace of W_{3}.

(41) There exists W such that the carrier of the carrier of W = (the carrier of the carrier of W_{1})∪ (the carrier of the carrier of W_{2}) if and only if W_{1} is a subspace of W_{2} or W_{2} is a subspace of W_{1}.

Let us consider G_{1}, V. The functor Subspaces V yielding a non-empty set is defined as follows: (Def.3) for every x holds x ∈ Subspaces V if and only if x is a subspace of V.

In the sequel D denotes a non-empty set. The following three propositions are true:

(42) If for every x holds x ∈ D if and only if x is a subspace of V, then D = Subspaces V.

(43) x ∈ Subspaces V if and only if x is a subspace of V.

(44) V ∈ Subspaces V.

Let us consider G_{1}, V, W_{1}, W_{2}. We say that V is the direct sum of W_{1} and W_{2} if and only if: (Def.4) V = W_{1} + W_{2} and W_{1} ∩ W_{2} = 0_{V}.

Let us consider G_{1}, V, W. A subspace of V is said to be a linear complement of W if: (Def.5) V is the direct sum of it and W.

We now state three propositions:

(45) V is the direct sum of W_{1} and W_{2} if and only if V = W_{1} + W_{2} and W_{1} ∩ W_{2} = 0_{V}.

(46) If V is the direct sum of W_{1} and W_{2}, then W_{1} is a linear complement of W_{2}.

(47) If V is the direct sum of W_{1} and W_{2}, then W_{2} is a linear complement

of W_{1}. In the sequel L denotes a linear complement of W. The following propositions

are true:

(48) V is the direct sum of L and W and V is the direct sum of W and L.

(49) W + L = V and L + W = V.

(50) W ∩ L = 0_{V} and L ∩ W = 0_{V}.

(51) If V is the direct sum of W_{1} and W_{2}, then V is the direct sum of W_{2} and W_{1}.

(52) V is the direct sum of 0_{V} and Ω_{V} and V is the direct sum of Ω_{V} and 0_{V}.

(53) W is a linear complement of L.

(54) 0_{V} is a linear complement of Ω_{V} and Ω_{V} is a linear complement of 0_{V}. In the sequel C_{1} is a coset of W_{1} and C_{2} is a coset of W_{2}. We now state

several propositions:

(55) If C_{1} ∩ C_{2} ≠ ∅, then C_{1} ∩ C_{2} is a coset of W_{1} ∩ W_{2}.

(56) V is the direct sum of W_{1} and W_{2} if and only if for every C_{1}, C_{2} there exists v such that C_{1} ∩ C_{2} = {v}.

(57) W_{1} + W_{2} = V if and only if for every v there exist v_{1}, v_{2} such that v_{1} ∈ W_{1} and v_{2} ∈ W_{2} and v = v_{1} + v_{2}.

(58) If V is the direct sum of W_{1} and W_{2} and v = v_{1} + v_{2} and v = u_{1} + u_{2} and v_{1} ∈ W_{1} and u_{1} ∈ W_{1} and v_{2} ∈ W_{2} and u_{2} ∈ W_{2}, then v_{1} = u_{1} and v_{2} = u_{2}.

(59) Suppose V = W_{1} + W_{2} and there exists v such that for all v_{1}, v_{2}, u_{1}, u_{2} such that v = v_{1} + v_{2} and v = u_{1} + u_{2} and v_{1} ∈ W_{1} and u_{1} ∈ W_{1} and v_{2} ∈ W_{2} and u_{2} ∈ W_{2} holds v_{1} = u_{1} and v_{2} = u_{2}. Then V is the direct sum of W_{1} and W_{2}.

In the sequel t will denote an element of [: the carrier of the carrier of V, the carrier of the carrier of V :]. Let us consider G_{1}, V, t. Then t_{1} is a vector of V. Then t_{2} is a vector of V.

Let us consider G_{1}, V, v, W_{1}, W_{2}. Let us assume that V is the direct sum of W_{1} and W_{2}. The functor v < (W_{1}, W_{2}) yielding an element of [: the carrier of the carrier of V, the carrier of the carrier of V :] is defined by:

(Def.6) v = (v < (W_{1}, W_{2}))_{1} + (v < (W_{1}, W_{2}))_{2} and (v < (W_{1}, W_{2}))_{1} ∈ W_{1} and (v < (W_{1}, W_{2}))_{2} ∈ W_{2}.

Next we state a number of propositions:

(60) If V is the direct sum of W_{1} and W_{2} and t_{1} + t_{2} = v and t_{1} ∈ W_{1} and t_{2} ∈ W_{2}, then t = v < (W_{1}, W_{2}).

(61) If V is the direct sum of W_{1} and W_{2}, then (v < (W_{1}, W_{2}))_{1} + (v < (W_{1}, W_{2}))_{2} = v.

(62) If V is the direct sum of W_{1} and W_{2}, then (v < (W_{1}, W_{2}))_{1} ∈ W_{1}.

(63) If V is the direct sum of W_{1} and W_{2}, then (v < (W_{1}, W_{2}))_{2} ∈ W_{2}.

(64) If V is the direct sum of W_{1} and W_{2}, then (v < (W_{1}, W_{2}))_{1} = (v < (W_{2}, W_{1}))_{2}.

(65) If V is the direct sum of W_{1} and W_{2}, then (v < (W_{1}, W_{2}))_{2} = (v < (W_{2}, W_{1}))_{1}.

(66) If t_{1} + t_{2} = v and t_{1} ∈ W and t_{2} ∈ L, then t = v < (W, L).

(67) (v < (W, L))_{1} + (v < (W, L))_{2} = v.

(68) (v < (W, L))_{1} ∈ W and (v < (W, L))_{2} ∈ L.

(69) (v < (W, L))_{1} = (v < (L, W))_{2}.

(70) (v < (W, L))_{2} = (v < (L, W))_{1}.

In the sequel A_{1}, A_{2} will be elements of Subspaces V. Let us consider G_{1}, V. The functor SubJoin V yields a binary operation on Subspaces V and is defined by:

(Def.7) for all A_{1}, A_{2}, W_{1}, W_{2} such that A_{1} = W_{1} and A_{2} = W_{2} holds (SubJoin V)(A_{1}, A_{2}) = W_{1} + W_{2}.

Let us consider G_{1}, V. The functor SubMeet V yielding a binary operation on Subspaces V is defined by: (Def.8) for all A_{1}, A_{2}, W_{1}, W_{2} such that A_{1} = W_{1} and A_{2} = W_{2} holds

(SubMeet V)(A_{1}, A_{2}) = W_{1} ∩ W_{2}.

In the sequel o denotes a binary operation on Subspaces V. One can prove the following propositions:

(71) If A_{1} = W_{1} and A_{2} = W_{2}, then SubJoin V (A_{1}, A_{2}) = W_{1} + W_{2}.

(72) If for all A_{1}, A_{2}, W_{1}, W_{2} such that A_{1} = W_{1} and A_{2} = W_{2} holds o(A_{1}, A_{2}) = W_{1} + W_{2}, then o = SubJoin V.

(73) If A_{1} = W_{1} and A_{2} = W_{2}, then SubMeet V (A_{1}, A_{2}) = W_{1} ∩ W_{2}.

(74) If for all A_{1}, A_{2}, W_{1}, W_{2} such that A_{1} = W_{1} and A_{2} = W_{2} holds o(A_{1}, A_{2}) = W_{1} ∩ W_{2}, then o = SubMeet V.

(75) ⟨Subspaces V, SubJoin V, SubMeet V ⟩ is a lattice.

(76) ⟨Subspaces V, SubJoin V, SubMeet V ⟩ is a lower bound lattice.

(77) ⟨Subspaces V, SubJoin V, SubMeet V ⟩ is an upper bound lattice.

(78) ⟨Subspaces V, SubJoin V, SubMeet V ⟩ is a bound lattice.

(79) ⟨Subspaces V, SubJoin V, SubMeet V ⟩ is a modular lattice.

(80) ⟨Subspaces V, SubJoin V, SubMeet V ⟩ is a complemented lattice.

(81) v = v_{1} + v_{2} if and only if v_{1} = v − v_{2}.

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175– 180, 1990.

[3] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[4] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[5] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[6] Wojciech A. Trybulec. Finite sums of vectors in vector space. Formalized Mathematics, 1(5):851–854, 1990.

[7] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.

876 Wojciech A. Trybulec

[8] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

[9] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathemat-

ics, 1(1):215–222, 1990.

Received July 27, 1990

Linear Combinations in Vector Space

Wojciech A. Trybulec^{1} Warsaw University

Summary. The notion of linear combination of vectors is introduced as a function from the carrier of a vector space to the carrier of the field. Definition of linear combination of set of vectors is also presented. We define addition and substraction of combinations and multiplication of combination by element of the field. Sum of finite set of vectors and sum of linear combination is defined. We prove theorems that belong rather to [5].

MML Identifier: VECTSP 6.

The articles [12], [4], [2], [1], [3], [11], [7], [6], [9], [5], [8], and [10] provide the terminology and notation for this paper. Let D be a non-empty set. Then ∅_{D} is a subset of D.

For simplicity we adopt the following rules: x will be arbitrary, i will be a natural number, G_{1} will be a field, V will be a vector space over G_{1}, u, v, v_{1}, v_{2}, v_{3} will be vectors of V, a, b, c will be elements of G_{1}, F, G will be finite sequences of elements of the carrier of the carrier of V, A, B will be subsets of V, and f will be a function from the carrier of the carrier of V into the carrier of G_{1}. Let us consider G_{1}, V. A subset of V is called a finite subset of V if: (Def.1) it is finite.

We now state the proposition

(1) A is a finite subset of V if and only if A is finite.

In the sequel S, T are finite subsets of V. Let us consider G_{1}, V, S, T. Then S ∪ T is a finite subset of V. Then S ∩ T is a finite subset of V. Then S \ T is a finite subset of V. Then S− . T is a finite subset of V.

Let us consider G_{1}, V. The functor 0_{V} yields a finite subset of V and is defined as follows: (Def.2) 0_{V} = ∅.

One can prove the following proposition (2) 0_{V} = ∅. Let us consider G_{1}, V, T. The functor ∑ T yields a vector of V and is defined

1Supported by RPBP.III-24.C1

as follows: (Def.3) there exists F such that rng F = T and F is one-to-one and ∑ T = ∑ F.

We now state two propositions:

(3) There exists F such that rng F = T and F is one-to-one and ∑ T = ∑ F.

(4) If rng F = T and F is one-to-one and v = ∑ F, then v = ∑ T. Let us consider G_{1}, V, v. Then {v} is a finite subset of V. Let us consider G_{1}, V, v_{1}, v_{2}. Then {v_{1}, v_{2}} is a finite subset of V. Let us consider G_{1}, V, v_{1}, v_{2}, v_{3}. Then {v_{1}, v_{2}, v_{3}} is a finite subset of V. One can prove the following propositions:

(5) ∑(0_{V}) = Θ_{V}.

(6) ∑{v} = v.

(7) If v_{1} ≠ v_{2}, then ∑{v_{1}, v_{2}} = v_{1} + v_{2}.

(8) If v_{1} ≠ v_{2} and v_{2} ≠ v_{3} and v_{1} ≠ v_{3}, then ∑{v_{1}, v_{2}, v_{3}} = (v_{1} + v_{2}) + v_{3}.

(9) If T misses S, then ∑(T ∪ S) = ∑ T + ∑ S.

(10) ∑(T ∪ S) = (∑ T + ∑ S) − ∑(T ∩ S).

(11) ∑(T ∩ S) = (∑ T + ∑ S) − ∑(T ∪ S).

(12) ∑(T \ S) = ∑(T ∪ S) − ∑ S.

(13) ∑(T \ S) = ∑ T − ∑(T ∩ S).

(14) ∑(T − . S) = ∑(T ∪ S) − ∑(T ∩ S).

(15) ∑(T − . S) = ∑(T \ S) + ∑(S \ T).

Let us consider G_{1}, V. An element of (the carrier of G_{1})^{the carrier of the carrier of V} is called a linear combination of V if:

(Def.4) there exists T such that for every v such that v ∉ T holds it(v) = 0_{G_{1}}.

In the sequel K, L, L_{1}, L_{2}, L_{3} are linear combinations of V. Next we state the proposition

(16) There exists T such that for every v such that v ∉ T holds L(v) = 0_{G_{1}}. In the sequel E is an element of (the carrier of G_{1})^{the carrier of the carrier of V}.

We now state the proposition

(17) If there exists T such that for every v such that v ∉ T holds E(v) = 0_{G_{1}},

then E is a linear combination of V. Let us consider G_{1}, V, L. The functor support L yields a finite subset of V

and is defined as follows: (Def.5) support L = {v : L(v) ≠ 0_{G_{1}}}.

The following propositions are true:

(18) support L = {v : L(v) ≠ 0_{G_{1}}}.

(19) x ∈ support L if and only if there exists v such that x = v and L(v) ≠ 0_{G_{1}}.

(20) L(v) = 0_{G_{1}} if and only if v ∉ support L.

Let us consider G_{1}, V. The functor 0_{LC_{V}} yielding a linear combination of V is defined as follows: (Def.6) support 0_{LC_{V}} = ∅.

Next we state two propositions:

(21) L = 0_{LC_{V}} if and only if support L = ∅.

(22) 0_{LC_{V}} (v) = 0_{G_{1}}.

Let us consider G_{1}, V, A. A linear combination of V is said to be a linear combination of A if: (Def.7) support it ⊆ A.

One can prove the following proposition

(23) If support L ⊆ A, then L is a linear combination of A.

In the sequel l denotes a linear combination of A. Next we state several propositions:

(24) support l ⊆ A.

(25) If A ⊆ B, then l is a linear combination of B.

(26) 0_{LC_{V}} is a linear combination of A.

(27) For every linear combination l of ∅_{the carrier of the carrier of V} holds l = 0_{LC_{V}}.

(28) L is a linear combination of support L.

Let us consider G_{1}, V, F, f. The functor f · F yields a finite sequence of elements of the carrier of the carrier of V and is defined by: (Def.8) len(f · F) = len F and for every i such that i ∈ dom(f · F) holds

(f · F)(i) = f(π_{i}F) · π_{i}F. Next we state several propositions:

(29) len(f · F) = len F.

(30) For every i such that i ∈ dom(f · F) holds (f · F)(i) = f(π_{i}F) · π_{i}F.

(31) If len G = len F and for every i such that i ∈ dom G holds G(i) = f(π_{i}F) · π_{i}F, then G = f · F.

(32) If i ∈ dom F and v = F (i), then (f · F)(i) = f(v) · v.

(33) f · ε_{the carrier of the carrier of V} = ε_{the carrier of the carrier of V}.

(34) f · ⟨v⟩ = ⟨f(v) · v⟩.

(35) f · ⟨v_{1}, v_{2}⟩ = ⟨f(v_{1}) · v_{1}, f(v_{2}) · v_{2}⟩.

(36) f · ⟨v_{1}, v_{2}, v_{3}⟩ = ⟨f(v_{1}) · v_{1}, f(v_{2}) · v_{2}, f(v_{3}) · v_{3}⟩.

(37) f · (F ^{}G) = (f · F)^{}(f · G).

Let us consider G_{1}, V, L. The functor ∑ L yielding a vector of V is defined as follows:

(Def.9) there exists F such that F is one-to-one and rng F = support L and

∑ L = ∑(L · F). The following propositions are true:

(38) There exists F such that F is one-to-one and rng F = support L and ∑ L = ∑(L · F).

(39) If F is one-to-one and rng F = support L and u = ∑(L · F), then u = ∑ L.

(40) A ≠ ∅ and A is linearly closed if and only if for every l holds ∑ l ∈ A.

(41) ∑ 0_{LC_{V}} = Θ_{V}.

(42) For every linear combination l of ∅_{the carrier of the carrier of V} holds ∑ l = Θ_{V}.

(43) For every linear combination l of {v} holds ∑ l = l(v) · v.

(44) If v_{1} ≠ v_{2}, then for every linear combination l of {v_{1}, v_{2}} holds ∑ l = l(v_{1}) · v_{1} + l(v_{2}) · v_{2}.

(45) If support L = ∅, then ∑ L = Θ_{V}.

(46) If support L = {v}, then ∑ L = L(v) · v.

(47) If support L = {v_{1}, v_{2}} and v_{1} ≠ v_{2}, then ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2}. Let us consider G_{1}, V, L_{1}, L_{2}. Let us note that one can characterize the

predicate L_{1} = L_{2} by the following (equivalent) condition: (Def.10) for every v holds L_{1}(v) = L_{2}(v).

One can prove the following proposition

(48) If for every v holds L_{1}(v) = L_{2}(v), then L_{1} = L_{2}.

Let us consider G_{1}, V, L_{1}, L_{2}. The functor L_{1}+L_{2} yields a linear combination of V and is defined as follows:

(Def.11) for every v holds (L_{1} + L_{2})(v) = L_{1}(v) + L_{2}(v).

Next we state several propositions:

(49) If for every v holds L(v) = L_{1}(v) + L_{2}(v), then L = L_{1} + L_{2}.

(50) (L_{1} + L_{2})(v) = L_{1}(v) + L_{2}(v).

(51) support(L_{1} + L_{2}) ⊆ support L_{1} ∪ support L_{2}.

(52) If L_{1} is a linear combination of A and L_{2} is a linear combination of A, then L_{1} + L_{2} is a linear combination of A.

(53) L_{1} + L_{2} = L_{2} + L_{1}.

(54) L_{1} + (L_{2} + L_{3}) = (L_{1} + L_{2}) + L_{3}.

(55) L + 0_{LC_{V}} = L and 0_{LC_{V}} + L = L.

Let us consider G_{1}, V, a, L. The functor a · L yielding a linear combination of V is defined by:

(Def.12) for every v holds (a · L)(v) = a · L(v).

The following propositions are true:

(56) If for every v holds K(v) = a · L(v), then K = a · L.

(57) (a · L)(v) = a · L(v).

(58) If a ≠ 0_{G_{1}}, then support(a · L) = support L.

(59) 0_{G_{1}} · L = 0_{LC_{V}}.

(60) If L is a linear combination of A, then a · L is a linear combination of A.

(61) (a + b) · L = a · L + b · L.

(62) a · (L_{1} + L_{2}) = a · L_{1} + a · L_{2}.

(63) a · (b · L) = (a · b) · L.

(64) (1_{G_{1}}) · L = L.

Let us consider G_{1}, V, L. The functor −L yields a linear combination of V and is defined by:

(Def.13) −L = (−1_{G_{1}}) · L.

The following propositions are true:

(65) −L = (−1_{G_{1}}) · L.

(66) (−L)(v) = −L(v).

(67) If L_{1} + L_{2} = 0_{LC_{V}}, then L_{2} = −L_{1}.

(68) support(−L) = support L.

(69) If L is a linear combination of A, then −L is a linear combination of A.

(70) −(−L) = L.

Let us consider G_{1}, V, L_{1}, L_{2}. The functor L_{1} − L_{2} yielding a linear combination of V is defined by:

(Def.14) L_{1} − L_{2} = L_{1} + (−L_{2}).

Next we state a number of propositions:

(71) L_{1} − L_{2} = L_{1} + (−L_{2}).

(72) (L_{1} − L_{2})(v) = L_{1}(v) − L_{2}(v).

(73) support(L_{1} − L_{2}) ⊆ support L_{1} ∪ support L_{2}.

(74) If L_{1} is a linear combination of A and L_{2} is a linear combination of A, then L_{1} − L_{2} is a linear combination of A.

(75) L − L = 0_{LC_{V}}.

(76) ∑(L_{1} + L_{2}) = ∑ L_{1} + ∑ L_{2}.

(77) ∑(a · L) = a · ∑ L.

(78) ∑(−L) = − ∑ L.

(79) ∑(L_{1} − L_{2}) = ∑ L_{1} − ∑ L_{2}.

(80) (−1_{G_{1}}) · a = −a.

(81) −1_{G_{1}} ≠ 0_{G_{1}}.

(82) −a = 0_{G_{1}} − a.

(83) −a = −(1_{G_{1}}) · a.

(84) (a − b) · c = a · c − b · c.

(85) If a + b = 0_{G_{1}}, then b = −a.

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[5] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[6] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

[7] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[8] Wojciech A. Trybulec. Finite sums of vectors in vector space. Formalized Mathematics, 1(5):851–854, 1990.

[9] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[10] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.

[11] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[12] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

Basis of Vector Space

Wojciech A. Trybulec^{1} Warsaw University

Summary. We prove the existence of a basis of a vector space, i.e., a set of vectors that generates the vector space and is linearly independent. We also introduce the notion of a subspace generated by a set of vectors and linear independence of set of vectors.

MML Identifier: VECTSP 7.

The terminology and notation used in this paper are introduced in the following papers: [5], [2], [9], [4], [3], [6], [1], [10], [8], and [7]. For simplicity we follow the rules: x will be arbitrary, G_{1} will denote a field, a, b will denote elements of G_{1}, V will denote a vector space over G_{1}, W will denote a subspace of V, v, v_{1}, v_{2} will denote vectors of V, A, B will denote subsets of V, and l will denote a linear combination of A. We now define two new predicates. Let us consider G_{1}, V, A. We say that A is linearly independent if and only if: (Def.1) for every l such that ∑ l = Θ_{V} holds support l = ∅. We say that A is linearly dependent if A is not linearly independent.

One can prove the following propositions:

(1) A is linearly independent if and only if for every l such that ∑ l = Θ_{V} holds support l = ∅.

(2) If A ⊆ B and B is linearly independent, then A is linearly independent.

(3) If A is linearly independent, then Θ_{V} ∉ A.

(4) ∅_{the carrier of the carrier of V} is linearly independent.

(5) {v} is linearly independent if and only if v ≠ Θ_{V}.

(6) If {v_{1}, v_{2}} is linearly independent, then v_{1} ≠ Θ_{V} and v_{2} ≠ Θ_{V}.

(7) {v, Θ_{V} } is linearly dependent and {Θ_{V} , v} is linearly dependent.

(8) v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent if and only if v_{2} ≠ Θ_{V} and for every a holds v_{1} ≠ a · v_{2}.

1Supported by RPBP.III-24.C1

(9) v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent if and only if for all a, b such that a · v_{1} + b · v_{2} = Θ_{V} holds a = 0_{G_{1}} and b = 0_{G_{1}}.

Let us consider G_{1}, V, A. The functor Lin(A) yields a subspace of V and is defined by: (Def.2) the carrier of the carrier of Lin(A) = {∑ l}.

The following propositions are true:

(10) If the carrier of the carrier of W = {∑ l}, then W = Lin(A).

(11) The carrier of the carrier of Lin(A) = {∑ l}.

(12) x ∈ Lin(A) if and only if there exists l such that x = ∑ l.

(13) If x ∈ A, then x ∈ Lin(A).

The following propositions are true:

(14) Lin(∅_{the carrier of the carrier of V}) = 0_{V}.

(15) If Lin(A) = 0_{V}, then A = ∅ or A = {Θ_{V} }.

(16) If A = the carrier of the carrier of W, then Lin(A) = W.

(17) If A = the carrier of the carrier of V, then Lin(A) = V.

(18) If A ⊆ B, then Lin(A) is a subspace of Lin(B).

(19) If Lin(A) = V and A ⊆ B, then Lin(B) = V.

(20) Lin(A ∪ B) = Lin(A) + Lin(B).

(21) Lin(A ∩ B) is a subspace of Lin(A) ∩ Lin(B).

(22) If A is linearly independent, then there exists B such that A ⊆ B and B is linearly independent and Lin(B) = V.

(23) If Lin(A) = V, then there exists B such that B ⊆ A and B is linearly

independent and Lin(B) = V. Let us consider G_{1}, V. A subset of V is called a basis of V if:

(Def.3) it is linearly independent and Lin(it) = V.

We now state the proposition

(24) If A is linearly independent and Lin(A) = V, then A is a basis of V. In the sequel I will denote a basis of V. We now state four propositions:

(25) I is linearly independent.

(26) Lin(I) = V.

(27) If A is linearly independent, then there exists I such that A ⊆ I.

(28) If Lin(A) = V, then there exists I such that I ⊆ A.

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[2] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[3] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[4] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[5] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[6] Wojciech A. Trybulec. Finite sums of vectors in vector space. Formalized Mathematics, 1(5):851–854, 1990.

[7] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877–882, 1990.

[8] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871–876, 1990.

[9] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

[10] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.

886

Factorial and Newton coeffitients

Rafał Kwiatek^{1} Nicolaus Copernicus University Toruń

Summary. We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that ∑_{k=0}^{n} \binom{n}{k} = 2^{n}.

MML Identifier: NEWTON.

The notation and terminology used in this paper have been introduced in the following articles: [4], [7], [6], [2], [3], [1], and [5]. We adopt the following rules: i, k, n, m, l denote natural numbers, a, b, x, y, z denote real numbers, and F, G denote finite sequences of elements of . One can prove the following propositions:

(1) For all x, y, z such that y ≠ 0 and z ≠ 0 holds \frac{z·x}{z·y} = \frac{x}{y}.

(2) If k ≥ l, then k − l is a natural number.

(3) For all F, G such that len F = len G and for every i such that i ∈ dom F holds F (i) = G(i) holds F = G.

(4) For every n such that n ≥ 1 holds 1 ∈ Seg n.

(5) For every n such that n ≥ 1 holds Seg n = ({1} ∪ {k : 1 < k ∧ k < n}) ∪ {n}.

(6) For every F holds len(a · F) = len F.

(7) n ∈ dom G if and only if n ∈ dom(a · G). Let us consider i, x. Then i ⟼ x is a finite sequence of elements of . Let us consider x, n. The functor x^{n} yielding a real number is defined as

follows: (Def.1) x^{n} = ∏(n ⟼ x).

One can prove the following propositions:

1Supported by RPBP.III-24.B5.

(8) x^{n} = ∏(n ⟼ x).

(9) For every x holds x^{0} = 1.

(10) For every x holds x^{1} = x.

(11) For every n holds x^{n+1} = x^{n} · x and x^{n+1} = x · x^{n}.

(12) (x · y)^{n} = x^{n} · y^{n}.

(13) x^{n+m} = x^{n} · x^{m}.

(14) (x^{n})^{m} = x^{n·m}.

(15) For every n holds 1^{n} = 1.

(16) For every n such that n ≥ 1 holds 0^{n} = 0.

Let us consider n. Then id_{n} is a finite sequence of elements of .

Let us consider x. Then ⟨x⟩ is a finite sequence of elements of . Let us consider y. Then ⟨x, y⟩ is a finite sequence of elements of .

Let us consider n. The functor n! yielding a real number is defined by: (Def.2) n! = ∏(id_{n}).

We now state several propositions:

(17) n! = ∏(id_{n}).

(18) 0! = 1.

(19) 1! = 1.

(20) 2! = 2.

(21) For every n holds (n + 1)! = (n + 1) · (n!) and (n + 1)! = (n!) · (n + 1).

(22) For every n holds n! is a natural number.

(23) For every n holds n! > 0.

(24) For every n holds n! ≠ 0.

(25) For all n, k holds (n!) · (k!) ≠ 0.

Let us consider k, n. The functor \binom{n}{k} yielding a real number is defined as follows: (Def.3) for every l such that l = n − k holds \binom{n}{k} = \frac{n!}{(k!)·(l!)} if n ≥ k, \binom{n}{k} = 0,

otherwise. We now state a number of propositions:

(26) For every l such that l = n − k holds \binom{n}{k} = \frac{n!}{(k!)·(l!)} if and only if n ≥ k or if \binom{n}{k} = 1, then n < k.

(27) \binom{0}{0} = 1.

(28) For every k such that k > 0 holds \binom{0}{k} = 0.

(29) For every n holds \binom{n}{0} = 1.

(30) For all n, k such that n ≥ k for every l such that l = n − k holds \binom{n}{k} = \binom{n}{l}.

(31) For every n holds \binom{n}{n} = 1.

(32) For all k, n such that k < n holds \binom{n+1}{k+1} = \binom{n}{k+1} + \binom{n}{k} and \binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.

(33) For every n such that n ≥ 1 holds \binom{n}{1} = n.

(34) For all n, l such that n ≥ 1 and l = n − 1 holds \binom{n}{l} = n.

(35) For every n and for every k holds \binom{n}{k} is a natural number.

(36) For all m, F such that m ≠ 0 and len F = m and for all i, l such that

i ∈ dom F and l = (n + i) − 1 holds F (i) = \binom{l}{n} holds ∑ F = \binom{n+m}{n+1}. Let a, b be real numbers, and let n be a natural number. The functor

⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩ yields a finite sequence of elements of and is defined as follows: (Def.4) len⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩ = n + 1 and for all i, l, m such that i ∈

dom⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩ and m = i − 1 and l = n − m holds ⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩(i) = (\binom{n}{m} · a^{l}) · b^{m}.

Next we state several propositions:

(37) Given F. Then the following conditions are equivalent:

(i) len F = n + 1 and for all i, l, m such that i ∈ dom F and m = i − 1 and l = n − m holds F (i) = (\binom{n}{m} · a^{l}) · b^{m},

(ii) F = ⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩.

(38) ⟨\binom{0}{0}a^{0}b^{0},…, \binom{0}{0}a^{0}b^{0}⟩ = ⟨1⟩.

(39) ⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩(1) = a^{n}.

(40) ⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩(n + 1) = b^{n}.

(41) For every n holds (a + b)^{n} = ∑⟨\binom{n}{0}a^{0}b^{n},…, \binom{n}{n}a^{n}b^{0}⟩.

Let us consider n. The functor ⟨\binom{n}{0},…, \binom{n}{n}⟩ yields a finite sequence of elements of and is defined by: (Def.5) len⟨\binom{n}{0},…, \binom{n}{n}⟩ = n + 1 and for all i, k such that i ∈ dom⟨\binom{n}{0},…, \binom{n}{n}⟩

and k = i − 1 holds ⟨\binom{n}{0},…, \binom{n}{n}⟩(i) = \binom{n}{k}. We now state three propositions:

(42) For every F holds len F = n + 1 and for all i, m such that i ∈ dom F and m = i − 1 holds F (i) = \binom{n}{m} if and only if F = ⟨\binom{n}{0},…, \binom{n}{n}⟩.

(43) For every n holds ⟨\binom{n}{0},…, \binom{n}{n}⟩ = ⟨\binom{n}{0}1^{0}1^{n},…, \binom{n}{n}1^{n}1^{0}⟩.

(44) For every n holds 2^{n} = ∑⟨\binom{n}{0},…, \binom{n}{n}⟩.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[5] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[7] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

Analytical Metric Affine Spaces and Planes ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. We introduce relations of orthogonality of vectors and of orthogonality of segments (considered as pairs of vectors) in real linear space of dimension two. This enables us to show an example of (in fact anisotropic and satisfying theorem on three perpendiculars) metric affine space (and plane as well). These two types of objects are defined formally as ”Mizar” modes. They are to be understood as structures consisting of a point universe and two binary relations on segments - a parallelity relation and orthogonality relation, satisfying appropriate axioms. With every such structure we correlate a structure obtained as a reduct of the given one to the parallelity relation only. Some relationships between metric affine spaces and their affine parts are proved; they enable us to use ”affine” facts and constructions in investigating metric affine geometry. We define the notions of line, parallelity of lines and two derived relations of orthogonality: between segments and lines, and between lines. Some basic properties of the introduced notions are proved.

MML Identifier: ANALMETR.

The articles [5], [1], [7], [6], [2], [3], and [4] provide the notation and terminology for this paper. For simplicity we follow a convention: V denotes a real linear space, u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, y denote vectors of V, a, a_{1}, a_{2}, b, b_{1}, b_{2} denote real numbers, and x, z are arbitrary. Let us consider V, w, y. We say that w, y span the space if and only if: (Def.1) for every u there exist a_{1}, a_{2} such that u = a_{1} · w + a_{2} · y and for all a_{1},

a_{2} such that a_{1} · w + a_{2} · y = 0_{V} holds a_{1} = 0 and a_{2} = 0. One can prove the following propositions:

(1) For all w, y holds w, y span the space if and only if for every u there exist a_{1}, a_{2} such that u = a_{1} · w + a_{2} · y and for all a_{1}, a_{2} such that a_{1} · w + a_{2} · y = 0_{V} holds a_{1} = 0 and a_{2} = 0.

1Supported by RPBP.III-24.C2.

(2) If w, y span the space, then there exist a_{1}, a_{2} such that u = a_{1}·w+a_{2}·y.

(3) If w, y span the space and a_{1} · w + a_{2} · y = 0_{V}, then a_{1} = 0 and a_{2} = 0. Let us consider V, u, v, w, y. We say that u, v are orthogonal w.r.t. w, y if

and only if: (Def.2) there exist a_{1}, a_{2}, b_{1}, b_{2} such that u = a_{1} ·w +a_{2} ·y and v = b_{1} ·w +b_{2} ·y

and a_{1} · b_{1} + a_{2} · b_{2} = 0. The following propositions are true:

(4) For all u, v, w, y holds u, v are orthogonal w.r.t. w, y if and only if there exist a_{1}, a_{2}, b_{1}, b_{2} such that u = a_{1} · w + a_{2} · y and v = b_{1} · w + b_{2} · y and a_{1} · b_{1} + a_{2} · b_{2} = 0.

(5) For all w, y such that w, y span the space holds u, v are orthogonal w.r.t. w, y if and only if for all a_{1}, a_{2}, b_{1}, b_{2} such that u = a_{1} · w + a_{2} · y and v = b_{1} · w + b_{2} · y holds a_{1} · b_{1} + a_{2} · b_{2} = 0.

(6) w, y are orthogonal w.r.t. w, y.

(7) There exists V and there exist w, y such that w, y span the space.

(8) If u, v are orthogonal w.r.t. w, y, then v, u are orthogonal w.r.t. w, y.

(9) If w, y span the space, then for all u, v holds u, 0_{V} are orthogonal w.r.t. w, y and 0_{V}, v are orthogonal w.r.t. w, y.

(10) If u, v are orthogonal w.r.t. w, y, then a · u, b · v are orthogonal w.r.t. w, y.

(11) If u, v are orthogonal w.r.t. w, y, then a · u, v are orthogonal w.r.t. w, y and u, b · v are orthogonal w.r.t. w, y.

(12) If w, y span the space, then for every u there exists v such that u, v are orthogonal w.r.t. w, y and v ≠ 0_{V}.

(13) If w, y span the space and v, u_{1} are orthogonal w.r.t. w, y and v, u_{2} are orthogonal w.r.t. w, y and v ≠ 0_{V}, then there exist a, b such that a · u_{1} = b · u_{2} but a ≠ 0 or b ≠ 0.

(14) If w, y span the space and u, v_{1} are orthogonal w.r.t. w, y and u, v_{2} are orthogonal w.r.t. w, y, then u, v_{1} + v_{2} are orthogonal w.r.t. w, y and u, v_{1} − v_{2} are orthogonal w.r.t. w, y.

(15) If w, y span the space and u, u are orthogonal w.r.t. w, y, then u = 0_{V}.

(16) If w, y span the space and u, u_{1} − u_{2} are orthogonal w.r.t. w, y and u_{1}, u_{2} − u are orthogonal w.r.t. w, y, then u_{2}, u − u_{1} are orthogonal w.r.t. w, y.

(17) If w, y span the space and u ≠ 0_{V}, then there exists a such that v−a·u, u are orthogonal w.r.t. w, y.

(18) u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1} or u, v \begin{matrix} \\ \\ ∥ \end{matrix} v_{1}, u_{1} if and only if there exist a, b such that a · (v − u) = b · (v_{1} − u_{1}) but a ≠ 0 or b ≠ 0.

(19) ⟨⟨⟨⟨u, v⟩⟩, ⟨⟨u_{1}, v_{1}⟩⟩⟩⟩ ∈ λ( _{V}) if and only if there exist a, b such that a · (v − u) = b · (v_{1} − u_{1}) but a ≠ 0 or b ≠ 0.

Let us consider V, u, u_{1}, v, v_{1}, w, y. We say that u, u_{1}, v and v_{1} are orthogonal w.r.t. w, y if and only if: (Def.3) u_{1} − u, v_{1} − v are orthogonal w.r.t. w, y.

One can prove the following proposition

(20) For all u, u_{1}, v, v_{1}, w, y holds u, u_{1}, v and v_{1} are orthogonal w.r.t. w,

y if and only if u_{1} − u, v_{1} − v are orthogonal w.r.t. w, y. Let us consider V, w, y. The ortogonality determined by w, y in V yielding a

binary relation on [: the vectors of V, the vectors of V :] is defined as follows: (Def.4) ⟨⟨x, z⟩⟩ ∈ the ortogonality determined by w, yinV if and only if there exist

u, u_{1}, v, v_{1} such that x = ⟨⟨u, u_{1}⟩⟩ and z = ⟨⟨v, v_{1}⟩⟩ and u, u_{1}, v and v_{1} are orthogonal w.r.t. w, y.

We now state the proposition

(21) For every binary relation R on [: the vectors of V, the vectors of V :] holds R = the ortogonality determined by w, yinV if and only if for all x, z holds ⟨⟨x, z⟩⟩ ∈ R if and only if there exist u, u_{1}, v, v_{1} such that x = ⟨⟨u, u_{1}⟩⟩ and z = ⟨⟨v, v_{1}⟩⟩ and u, u_{1}, v and v_{1} are orthogonal w.r.t. w, y.

In the sequel p, p_{1}, q, q_{1} will denote elements of the points of Λ(OASpace V). We now state three propositions:

(22) The points of Λ(OASpace V) = the vectors of V.

(23) The congruence of Λ(OASpace V) = λ( _{V}).

(24) If p = u and q = v and p_{1} = u_{1} and q_{1} = v_{1}, then p, q ∥ p_{1}, q_{1} if and only if there exist a, b such that a · (v − u) = b · (v_{1} − u_{1}) but a ≠ 0 or b ≠ 0.

We consider metric affine structures which are systems

⟨points, a parallelity, an orthogonality⟩, where the points constitute a non-empty set, the parallelity is a binary relation on [: the points, the points :], and the orthogonality is a binary relation on [: the points, the points :]. In the sequel P_{1} will denote a metric-affine structure. We now define two new predicates. Let us consider P_{1}, and let a, b, c, d be elements of the points of P_{1}. The predicate a, b ∥ c, d is defined as follows: (Def.5) ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the parallelity of P_{1}. The predicate a, b ⊥ c, d is defined as follows: (Def.6) ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the orthogonality of P_{1}.

One can prove the following propositions:

(25) For all elements a, b, c, d of the points of P_{1} holds a, b ∥ c, d if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the parallelity of P_{1}.

(26) For all elements a, b, c, d of the points of P_{1} holds a, b ⊥ c, d if and only

if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the orthogonality of P_{1}. Let us consider V, w, y. Let us assume that w, y span the space. The functor

AMSp(V, w, y) yielding a metric-affine structure is defined by:

(Def.7) AMSp(V, w, y) = ⟨ the vectors of

V, λ( _{V}), the ortogonality determined by w, y in V ⟩. Next we state two propositions:

(27) If w, y span the space, then P_{1} = AMSp(V, w, y) if and only if P_{1} = ⟨ the vectors of V, λ( _{V}), the ortogonality determined by w, y in V ⟩.

(28) If w, y span the space, then the points of AMSp(V, w, y) = the vectors of V and the parallelity of AMSp(V, w, y) = λ( _{V}) and the orthogonality of AMSp(V, w, y) = the ortogonality determined by w, y in V.

Let us consider P_{1}. The affine reduct of P_{1} yielding an affine structure is defined by: (Def.8) the affine reduct of P_{1} = ⟨ the points of P_{1}, the parallelity of P_{1}⟩.

We now state two propositions:

(29) For every P_{1} and for every A_{1} being an affine structure holds A_{1} = the affine reduct of P_{1} if and only if A_{1} = ⟨ the points of P_{1}, the parallelity of P_{1}⟩.

(30) If w, y span the space, then

the affine reduct of AMSp(V, w, y) = Λ(OASpace V). In the sequel p, p_{1}, p_{2}, q, q_{1}, r, r_{1}, r_{2} denote elements of the points of

AMSp(V, w, y). One can prove the following propositions:

(31) If w, y span the space and p = u and p_{1} = u_{1} and q = v and q_{1} = v_{1}, then p, q ⊥ p_{1}, q_{1} if and only if u, v, u_{1} and v_{1} are orthogonal w.r.t. w, y.

(32) If w, y span the space and p = u and q = v and p_{1} = u_{1} and q_{1} = v_{1}, then p, q ∥ p_{1}, q_{1} if and only if there exist a, b such that a · (v − u) = b · (v_{1} − u_{1}) but a ≠ 0 or b ≠ 0.

(33) If w, y span the space and p, q ⊥ p_{1}, q_{1}, then p_{1}, q_{1} ⊥ p, q.

(34) If w, y span the space and p, q ⊥ p_{1}, q_{1}, then p, q ⊥ q_{1}, p_{1}.

(35) If w, y span the space, then for all p, q, r holds p, q ⊥ r, r.

(36) If w, y span the space and p, p_{1} ⊥ q, q_{1} and p, p_{1} ∥ r, r_{1}, then p = p_{1} or q, q_{1} ⊥ r, r_{1}.

(37) If w, y span the space, then for every p, q, r there exists r_{1} such that p, q ⊥ r, r_{1} and r ≠ r_{1}.

(38) If w, y span the space and p, p_{1} ⊥ q, q_{1} and p, p_{1} ⊥ r, r_{1}, then p = p_{1} or q, q_{1} ∥ r, r_{1}.

(39) If w, y span the space and p, q ⊥ r, r_{1} and p, q ⊥ r, r_{2}, then p, q ⊥ r_{1}, r_{2}.

(40) If w, y span the space and p, q ⊥ p, q, then p = q.

(41) If w, y span the space and p, q ⊥ p_{1}, p_{2} and p_{1}, q ⊥ p_{2}, p, then p_{2}, q ⊥ p, p_{1}.

(42) If w, y span the space and p ≠ p_{1}, then for every q there exists q_{1} such

that p, p_{1} ∥ p, q_{1} and p, p_{1} ⊥ q_{1}, q. A metric-affine structure is called a metric affine space if:

(Def.9) (i) ⟨ the points of it, the parallelity of it⟩ is an affine space,

(ii) for all elements a, b, c, d, p, q, r, s of the points of it holds if a, b ⊥ a, b, then a = b but a, b ⊥ c, c but if a, b ⊥ c, d, then a, b ⊥ d, c and c, d ⊥ a, b but if a, b ⊥ p, q and a, b ∥ r, s, then p, q ⊥ r, s or a = b but if a, b ⊥ p, q and a, b ⊥ p, s, then a, b ⊥ q, s,

(iii) for all elements a, b, c of the points of it such that a ≠ b there exists

an element x of the points of it such that a, b ∥ a, x and a, b ⊥ x, c, (iv) for every elements a, b, c of the points of it there exists an element x

of the points of it such that a, b ⊥ c, x and c ≠ x.

We now state two propositions:

(43) Given P_{1}. Then P_{1} is a metric affine space if and only if the following

conditions are satisfied:

(i) ⟨ the points of P_{1}, the parallelity of P_{1}⟩ is an affine space, (ii) for all elements a, b, c, d, p, q, r, s of the points of P_{1} holds if a, b ⊥ a, b,

then a = b but a, b ⊥ c, c but if a, b ⊥ c, d, then a, b ⊥ d, c and c, d ⊥ a, b but if a, b ⊥ p, q and a, b ∥ r, s, then p, q ⊥ r, s or a = b but if a, b ⊥ p, q and a, b ⊥ p, s, then a, b ⊥ q, s,

(iii) for all elements a, b, c of the points of P_{1} such that a ≠ b there exists

an element x of the points of P_{1} such that a, b ∥ a, x and a, b ⊥ x, c, (iv) for every elements a, b, c of the points of P_{1} there exists an element x

of the points of P_{1} such that a, b ⊥ c, x and c ≠ x.

(44) If w, y span the space, then AMSp(V, w, y) is a metric affine space.

A metric-affine structure is said to be a metric affine plane if: (Def.10) (i) ⟨ the points of it, the parallelity of it⟩ is an affine plane,

(ii) for all elements a, b, c, d, p, q, r, s of the points of it holds if a, b ⊥ a, b, then a = b but a, b ⊥ c, c but if a, b ⊥ c, d, then a, b ⊥ d, c and c, d ⊥ a, b but if a, b ⊥ p, q and a, b ∥ r, s, then p, q ⊥ r, s or a = b but if a, b ⊥ p, q and a, b ⊥ r, s, then p, q ∥ r, s or a = b,

(iii) for every elements a, b, c of the points of it there exists an element x of the points of it such that a, b ⊥ c, x and c ≠ x.

Next we state four propositions:

(45) Given P_{1}. Then P_{1} is a metric affine plane if and only if the following

conditions are satisfied:

(i) ⟨ the points of P_{1}, the parallelity of P_{1}⟩ is an affine plane, (ii) for all elements a, b, c, d, p, q, r, s of the points of P_{1} holds if a, b ⊥ a, b,

then a = b but a, b ⊥ c, c but if a, b ⊥ c, d, then a, b ⊥ d, c and c, d ⊥ a, b but if a, b ⊥ p, q and a, b ∥ r, s, then p, q ⊥ r, s or a = b but if a, b ⊥ p, q and a, b ⊥ r, s, then p, q ∥ r, s or a = b,

(iii) for every elements a, b, c of the points of P_{1} there exists an element x

of the points of P_{1} such that a, b ⊥ c, x and c ≠ x.

(46) If w, y span the space, then AMSp(V, w, y) is a metric affine plane.

(47) For an arbitrary x holds x is an element of the points of P_{1} if and only if x is an element of the points of the affine reduct of P_{1}.

(48) For all elements a, b, c, d of the points of P_{1} and for all elements a′, b′, c′, d′ of the points of the affine reduct of P_{1} such that a = a′ and b = b′ and c = c′ and d = d′ holds a, b ∥ c, d if and only if a′, b′ ∥ c′, d′.

Let P_{1} be a metric affine space. Then the affine reduct of P_{1} is an affine space. Let P_{1} be a metric affine plane. Then the affine reduct of P_{1} is an affine plane. The following proposition is true

(49) For every metric affine plane P_{1} holds P_{1} is a metric affine space. We see that the metric affine plane is a metric affine space. The following two propositions are true:

(50) For every metric affine space P_{1} such that the affine reduct of P_{1} is an affine plane holds P_{1} is a metric affine plane.

(51) Let P_{1} be a metric-affine structure. Then P_{1} is a metric affine plane if

and only if the following conditions are satisfied:

(i) there exist elements a, b of the points of P_{1} such that a ≠ b, (ii) for all elements a, b, c, d, p, q, r, s of the points of P_{1} holds a, b ∥ b, a

and a, b ∥ c, c but if a, b ∥ p, q and a, b ∥ r, s, then p, q ∥ r, s or a = b but if a, b ∥ a, c, then b, a ∥ b, c and there exists an element x of the points of P_{1} such that a, b ∥ c, x and a, c ∥ b, x and there exist elements x, y, z of the points of P_{1} such that x, y x, z and there exists an element x of the points of P_{1} such that a, b ∥ c, x and c ≠ x but if a, b ∥ b, d and b ≠ a, then there exists an element x of the points of P_{1} such that c, b ∥ b, x and c, a ∥ d, x but if a, b ⊥ a, b, then a = b and a, b ⊥ c, c but if a, b ⊥ c, d, then a, b ⊥ d, c and c, d ⊥ a, b but if a, b ⊥ p, q and a, b ∥ r, s, then p, q ⊥ r, s or a = b but if a, b ⊥ p, q and a, b ⊥ r, s, then p, q ∥ r, s or a = b and there exists an element x of the points of P_{1} such that a, b ⊥ c, x and c ≠ x but if a, b c, d, then there exists an element x of the points of P_{1} such that a, b ∥ a, x and c, d ∥ c, x.

In the sequel x, a, b, c, d, p, q will denote elements of the points of P_{1}. Let us consider P_{1}, a, b, c. The predicate L(a, b, c) is defined as follows:

(Def.11) a, b ∥ a, c.

We now state the proposition

(52) For every P_{1} and for all a, b, c holds L(a, b, c) if and only if a, b ∥ a, c. Let us consider P_{1}, a, b. The functor Line(a, b) yielding a subset of the points

of P_{1} is defined by: (Def.12) for every element x of the points of P_{1} holds x ∈ Line(a, b) if and only

if L(a, b, x).

In the sequel A, K, M denote subsets of the points of P_{1}. The following proposition is true

(53) A = Line(a, b) if and only if for every x holds x ∈ A if and only if

L(a, b, x). Let us consider P_{1}, A. We say that A is a line if and only if:

(Def.13) there exist a, b such that a ≠ b and A = Line(a, b).

Next we state several propositions:

(54) A is a line if and only if there exist a, b such that a ≠ b and A = Line(a, b).

(55) For every metric affine space P_{1} and for all elements a, b, c of the points of P_{1} and for all elements a′, b′, c′ of the points of the affine reduct of P_{1} such that a = a′ and b = b′ and c = c′ holds L(a, b, c) if and only if L(a′, b′, c′).

(56) For every metric affine space P_{1} and for all elements a, b of the points of P_{1} and for all elements a′, b′ of the points of the affine reduct of P_{1} such that a = a′ and b = b′ holds Line(a, b) = Line(a′, b′).

(57) For an arbitrary X holds X is a subset of the points of P_{1} if and only if X is a subset of the points of the affine reduct of P_{1}.

(58) For every metric affine space P_{1} and for every subset X of the points of P_{1} and for every subset Y of the points of the affine reduct of P_{1} such that X = Y holds X is a line if and only if Y is a line.

Let us consider P_{1}, a, b, K. The predicate a, b ⊥ K is defined as follows: (Def.14) there exist p, q such that p ≠ q and K = Line(p, q) and a, b ⊥ p, q.

Let us consider P_{1}, K, M. The predicate K ⊥ M is defined by: (Def.15) there exist p, q such that p ≠ q and K = Line(p, q) and p, q ⊥ M.

Let us consider P_{1}, K, M. The predicate K ∥ M is defined by: (Def.16) there exist a, b, c, d such that a ≠ b and c ≠ d and K = Line(a, b) and

M = Line(c, d) and a, b ∥ c, d.

One can prove the following propositions:

(59) For all a, b, K holds a, b ⊥ K if and only if there exist p, q such that p ≠ q and K = Line(p, q) and a, b ⊥ p, q.

(60) For all K, M holds K ⊥ M if and only if there exist p, q such that p ≠ q and K = Line(p, q) and p, q ⊥ M.

(61) For all K, M holds K ∥ M if and only if there exist a, b, c, d such that a ≠ b and c ≠ d and K = Line(a, b) and M = Line(c, d) and a, b ∥ c, d.

(62) If a, b ⊥ K, then K is a line but if K ⊥ M, then K is a line and M is a line.

(63) K ⊥ M if and only if there exist a, b, c, d such that a ≠ b and c ≠ d and K = Line(a, b) and M = Line(c, d) and a, b ⊥ c, d.

(64) For every metric affine space P_{1} and for all subsets M, N of the points of P_{1} and for all subsets M′, N′ of the points of the affine reduct of P_{1} such that M = M′ and N = N′ holds M ∥ N if and only if M′ ∥ N′.

We adopt the following rules: P_{1} denotes a metric affine space, A, K, M, N denote subsets of the points of P_{1}, and a, b, c, d, p, q, r, s denote elements of the points of P_{1}. The following propositions are true:

(65) If K is a line, then a, a ⊥ K.

(66) If a, b ⊥ K but a, b ∥ c, d or c, d ∥ a, b and a ≠ b, then c, d ⊥ K.

(67) If a, b ⊥ K, then b, a ⊥ K.

(68) If K ∥ M, then M ∥ K.

(69) If r, s ⊥ K but K ∥ M or M ∥ K, then r, s ⊥ M.

(70) If K ⊥ M, then M ⊥ K.

(71) If a ∈ K and b ∈ K and a, b ⊥ K, then a = b.

(72) If K is a line, then K ⊥̸ K.

(73) If K ⊥ M or M ⊥ K but K ∥ N or N ∥ K, then M ⊥ N and N ⊥ M.

(74) If K ∥ N, then K ⊥̸ N.

(75) If a ∈ K and b ∈ K and c, d ⊥ K, then c, d ⊥ a, b and a, b ⊥ c, d.

(76) If a ∈ K and b ∈ K and a ≠ b and K is a line, then K = Line(a, b).

(77) If a ∈ K and b ∈ K and a ≠ b and K is a line but a, b ⊥ c, d or c, d ⊥ a, b, then c, d ⊥ K.

(78) If a ∈ M and b ∈ M and c ∈ N and d ∈ N and M ⊥ N, then a, b ⊥ c, d.

(79) If p ∈ M and p ∈ N and a ∈ M and b ∈ N and a ≠ b and a ∈ K and b ∈ K and A ⊥ M and A ⊥ N and K is a line, then A ⊥ K.

(80) b, c ⊥ a, a and a, a ⊥ b, c and b, c ∥ a, a and a, a ∥ b, c.

(81) If a, b ∥ c, d, then a, b ∥ d, c and b, a ∥ c, d and b, a ∥ d, c and c, d ∥ a, b and c, d ∥ b, a and d, c ∥ a, b and d, c ∥ b, a.

(82) Suppose that (i) p ≠ q, (ii) p, q ∥ a, b and p, q ∥ c, d or p, q ∥ a, b and c, d ∥ p, q or a, b ∥ p, q and

c, d ∥ p, q or a, b ∥ p, q and p, q ∥ c, d.

Then a, b ∥ c, d.

(83) If a, b ⊥ c, d, then a, b ⊥ d, c and b, a ⊥ c, d and b, a ⊥ d, c and c, d ⊥ a, b and c, d ⊥ b, a and d, c ⊥ a, b and d, c ⊥ b, a.

(84) Suppose that (i) p ≠ q, (ii) p, q ∥ a, b and p, q ⊥ c, d or p, q ∥ c, d and p, q ⊥ a, b or p, q ∥ a, b and

c, d ⊥ p, q or p, q ∥ c, d and a, b ⊥ p, q or a, b ∥ p, q and c, d ⊥ p, q or c, d ∥ p, q and a, b ⊥ p, q or a, b ∥ p, q and p, q ⊥ c, d or c, d ∥ p, q and p, q ⊥ a, b. Then a, b ⊥ c, d.

We follow the rules: P_{1} is a metric affine plane, K, M, N are subsets of the points of P_{1}, and x, a, b, c, d, p, q are elements of the points of P_{1}. The following propositions are true:

(85) Suppose that (i) p ≠ q, (ii) p, q ⊥ a, b and p, q ⊥ c, d or p, q ⊥ a, b and c, d ⊥ p, q or a, b ⊥ p, q and

c, d ⊥ p, q or a, b ⊥ p, q and p, q ⊥ c, d.

Then a, b ∥ c, d.

(86) If a ∈ M and b ∈ M and a ≠ b and M is a line and c ∈ N and d ∈ N and c ≠ d and N is a line and a, b ∥ c, d, then M ∥ N.

(87) If K ⊥ M or M ⊥ K but K ⊥ N or N ⊥ K, then M ∥ N and N ∥ M.

(88) If M ⊥ N, then there exists p such that p ∈ M and p ∈ N.

(89) If a, b ⊥ c, d, then there exists p such that L(a, b, p) and L(c, d, p).

(90) If a, b ⊥ K, then there exists p such that L(a, b, p) and p ∈ K.

(91) There exists x such that a, x ⊥ p, q and L(p, q, x).

(92) If K is a line, then there exists x such that a, x ⊥ K and x ∈ K.

References

[1] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[5] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[6] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[7] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

900

Projective Spaces - part II

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. Distinction is made among several types of many dimensional projective spaces - at least three dimensional and exactly threedimensional projective structures. We prove that analytical projective spaces defined over appropiate real linear spaces may serve as examples of the introduced classes of projective spaces. Corresponding subclasses of Fano projective structures are distinguished. Note that in projective geometry the axiom which assures that the dimension is not greater than three can be formulated as the statement: there exists a plane which intersects every line.

MML Identifier: ANPROJ 3.

The terminology and notation used in this paper have been introduced in the following articles: [1], [4], [2], and [3]. We follow a convention: V will be a real linear space, p, q, r, s, u, v, w, y, u_{1}, v_{1} will be vectors of V, and a, b, c, d, a_{1}, b_{1}, c_{1} will be real numbers. The following two propositions are true:

(1) Suppose that

(i) for every w there exist a, b, c, d such that w = ((a·p+b·q)+c·r)+d·s, (ii) for all a, b, c, d such that ((a · p + b · q) + c · r) + d · s = 0_{V} holds a = 0

and b = 0 and c = 0 and d = 0. Then for all u, v such that u is a proper vector and v is a proper vector there exist y, w such that u, v and w are lineary dependent and q, r and y are lineary dependent and p, w and y are lineary dependent and y is a proper vector and w is a proper vector.

(2) Suppose for all a, b, a_{1}, b_{1} such that ((a·u+b·v)+a_{1} ·u_{1})+b_{1} ·v_{1} = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0. Then for no y holds y is a proper vector and u, v and y are lineary dependent and u_{1}, v_{1} and y are lineary dependent.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

We adopt the following rules: V will be a non-trivial real linear space, u, v, w, y, w_{1} will be vectors of V, and p, p_{1}, q, q_{1}, q_{2}, q_{3}, r, r_{1}, r_{2}, r_{3} will be elements of the points of the projective space over V. We now state two propositions:

(3) If there exist p, q, r such that p, q and r are not collinear, then for all p, q such that p ≠ q there exists r such that p, q and r are not collinear.

(4) Suppose that

(i) there exist y, u, v, w such that for every w_{1} there exist a, b, a_{1}, b_{1} such that w_{1} = ((a · y + b · u) + a_{1} · v) + b_{1} · w and for all a, b, a_{1}, b_{1} such that ((a · y + b · u) + a_{1} · v) + b_{1} · w = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0. Then there exist p, q_{1}, q_{2} such that p, q_{1} and q_{2} are not collinear and for every r_{1}, r_{2} there exist q_{3}, r_{3} such that r_{1}, r_{2} and r_{3} are collinear and q_{1}, q_{2} and q_{3} are collinear and p, r_{3} and q_{3} are collinear.

Next we state the proposition

(5) Suppose that

(i) there exist p, q, r such that p, q and r are not collinear, (ii) for every p, q there exists r such that p ≠ r and q ≠ r and p, q and r

are collinear, (iii) there exist p, q_{1}, q_{2} such that p, q_{1} and q_{2} are not collinear and for

every r_{1}, r_{2} there exist q_{3}, r_{3} such that r_{1}, r_{2} and r_{3} are collinear and q_{1}, q_{2} and q_{3} are collinear and p, r_{3} and q_{3} are collinear. Then for every p, p_{1}, q, q_{1}, r_{2} there exist r, r_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

In the sequel u, v, w, y, u_{1}, v_{1}, w_{1} will be vectors of V. Next we state three propositions:

(6) Suppose that

(i) there exist y, u, v, w such that for every w_{1} there exist a, b, c, c_{1} such that w_{1} = ((a · y + b · u) + c · v) + c_{1} · w and for all a, b, a_{1}, b_{1} such that ((a · y + b · u) + a_{1} · v) + b_{1} · w = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0. Then for every p, p_{1}, q, q_{1}, r_{2} there exist r, r_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

(7) Suppose there exist u, v, u_{1}, v_{1} such that for all a, b, a_{1}, b_{1} such that ((a · u + b · v) + a_{1} · u_{1}) + b_{1} · v_{1} = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0. Then there exist p, p_{1}, q, q_{1} such that for no r holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(8) Suppose that

(i) there exist u, v, u_{1}, v_{1} such that for every w there exist a, b, a_{1}, b_{1} such that w = ((a · u + b · v) + a_{1} · u_{1}) + b_{1} · v_{1} and for all a, b, a_{1}, b_{1} such that ((a · u + b · v) + a_{1} · u_{1}) + b_{1} · v_{1} = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0. Then (ii) for every p, p_{1}, q, q_{1}, r_{2} there exist r, r_{1} such that p, q and r are

collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear, (iii) for every p, q there exists r such that p ≠ r and q ≠ r and p, q and r

are collinear, (iv) there exist p, q, r such that p, q and r are not collinear,

(v) there exist p, p_{1}, q, q_{1} such that for no r holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A projective space defined in terms of collinearity is called an at least 3 dimensional projective space defined in terms of collinearity if:

(Def.1) there exist elements p, p_{1}, q, q_{1} of the points of it such that for no element r of the points of it holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

We now state three propositions:

(9) For every projective space C_{1} defined in terms of collinearity holds C_{1} is an at least 3 dimensional projective space defined in terms of collinearity if and only if there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(10) If there exist u, v, u_{1}, v_{1} such that for all a, b, a_{1}, b_{1} such that ((a · u + b · v) + a_{1} · u_{1}) + b_{1} · v_{1} = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0, then the projective space over V is an at least 3 dimensional projective space defined in terms of collinearity.

(11) Let C_{1} be a collinearity structure. Then C_{1} is an at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

An at least 3 dimensional projective space defined in terms of collinearity is said to be a Fanoian at least 3 dimensional projective space defined in terms of collinearity if:

(Def.2) Let p_{1}, r_{2}, q, r_{1}, q_{1}, p, r be elements of the points of it . Suppose p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear. Then p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

One can prove the following propositions:

(12) Let C_{1} be an at least 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Fanoian at least 3 dimensional projective space defined in terms of collinearity if and only if for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(13) If there exist u, v, u_{1}, v_{1} such that for all a, b, a_{1}, b_{1} such that ((a · u + b · v) + a_{1} · u_{1}) + b_{1} · v_{1} = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0, then the projective space over V is a Fanoian at least 3 dimensional projective space defined in terms of collinearity.

(14) Let C_{1} be a collinearity structure. Then C_{1} is a Fanoian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vi) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(15) For every C_{1} being a collinearity structure holds C_{1} is a Fanoian at least 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Fanoian projective space defined in terms of collinearity and there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

An at least 3 dimensional projective space defined in terms of collinearity is called a 3 dimensional projective space defined in terms of collinearity if: (Def.3) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of it there exist elements

r, r_{1} of the points of it such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

The following propositions are true:

(16) For every at least 3 dimensional projective space C_{1} defined in terms of collinearity holds C_{1} is a 3 dimensional projective space defined in terms of collinearity if and only if for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

(17) Suppose that

(i) there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}. Then the projective space over V is a 3 dimensional projective space defined in terms of collinearity.

(18) Let C_{1} be a collinearity structure. Then C_{1} is a 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

A 3 dimensional projective space defined in terms of collinearity is called a Fanoian 3 dimensional projective space defined in terms of collinearity if:

(Def.4) Let p_{1}, r_{2}, q, r_{1}, q_{1}, p, r be elements of the points of it . Suppose p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear. Then p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

We now state four propositions:

(19) Let C_{1} be a 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Fanoian 3 dimensional projective space defined in terms of collinearity if and only if for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(20) Suppose that

(i) there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}. Then the projective space over V is a Fanoian 3 dimensional projective space defined in terms of collinearity.

(21) Let C_{1} be a collinearity structure. Then C_{1} is a Fanoian 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vi) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear, (vii) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements

r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

(22) For every C_{1} being a collinearity structure holds C_{1} is a Fanoian 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Fanoian at least 3 dimensional projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.

[3] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part I. Formalized Mathematics, 1(4):767–776, 1990.

[4] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.

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Projective Spaces - part III

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. In the classes of projective spaces, defined in terms of collinearity, introduced in the article [3], we distinguish the subclasses of Desarguesian projective structures. As examples of these types of objects we consider analytical projective spaces defined over suitable real linear spaces; analytical counterpart of the Desargues Axiom is proved without any assumption on the dimension of the underlying linear space.

MML Identifier: ANPROJ 4.

The articles [1], [4], [2], and [3] provide the notation and terminology for this paper. We adopt the following rules: V will denote a real linear space, o, p, p_{1}, p_{2}, p_{3}, q, q_{1}, q_{2}, q_{3}, r, r_{1}, r_{2}, r_{3} will denote vectors of V, and a, b, c, a_{1}, b_{1}, a_{2}, c_{2} will denote real numbers. Let us consider V, p_{1}, p_{2}, p_{3}. We say that p_{1}, p_{2} and p_{3} are proper vectors if and only if: (Def.1) p_{1} is a proper vector and p_{2} is a proper vector and p_{3} is a proper vector.

Next we state the proposition

(1) p_{1}, p_{2} and p_{3} are proper vectors if and only if p_{1} is a proper vector and p_{2} is a proper vector and p_{3} is a proper vector.

Let us consider V, p_{1}, p_{2}, p_{3}, r_{1}, r_{2}, r_{3}. We say that p_{1}, p_{2}, p_{3}, r_{1}, r_{2}, and r_{3} lie on a triangle if and only if: (Def.2) p_{1}, p_{2} and r_{3} are lineary dependent and p_{1}, p_{3} and r_{2} are lineary de-

pendent and p_{2}, p_{3} and r_{1} are lineary dependent.

Next we state the proposition

(2) p_{1}, p_{2}, p_{3}, r_{1}, r_{2}, and r_{3} lie on a triangle if and only if p_{1}, p_{2} and r_{3} are lineary dependent and p_{1}, p_{3} and r_{2} are lineary dependent and p_{2}, p_{3} and r_{1} are lineary dependent.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

Let us consider V, o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}. We say that o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, and q_{3} are perspective if and only if: (Def.3) o, p_{1} and q_{1} are lineary dependent and o, p_{2} and q_{2} are lineary dependent

and o, p_{3} and q_{3} are lineary dependent.

The following propositions are true:

(3) o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, and q_{3} are perspective if and only if o, p_{1} and q_{1} are lineary dependent and o, p_{2} and q_{2} are lineary dependent and o, p_{3} and q_{3} are lineary dependent.

(4) Suppose o, p_{1} and q_{1} are lineary dependent and o and p_{1} are not proportional and o and q_{1} are not proportional and p_{1} and q_{1} are not proportional and o, p_{1} and q_{1} are proper vectors. Then there exist a_{1}, b_{1} such that b_{1} · q_{1} = o + a_{1} · p_{1} and a_{1} ≠ 0 and b_{1} ≠ 0 and there exist a_{2}, c_{2} such that q_{1} = c_{2} · o + a_{2} · p_{1} and c_{2} ≠ 0 and a_{2} ≠ 0.

(5) If p, q and r are lineary dependent and p and q are not proportional and p, q and r are proper vectors, then there exist a, b such that r = a·p+b·q.

(6) Suppose that

(i) o is a proper vector, (ii) p_{1}, p_{2} and p_{3} are proper vectors, (iii) q_{1}, q_{2} and q_{3} are proper vectors, (iv) r_{1}, r_{2} and r_{3} are proper vectors,

(v) o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, and q_{3} are perspective, (vi) o and q_{1} are not proportional, (vii) o and q_{2} are not proportional, (viii) o and q_{3} are not proportional, (ix) p_{1} and q_{1} are not proportional,

(x) p_{2} and q_{2} are not proportional, (xi) p_{3} and q_{3} are not proportional, (xii) o, p_{1} and p_{2} are not lineary dependent, (xiii) o, p_{1} and p_{3} are not lineary dependent, (xiv) o, p_{2} and p_{3} are not lineary dependent, (xv) p_{1}, p_{2}, p_{3}, r_{1}, r_{2}, and r_{3} lie on a triangle, (xvi) q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, and r_{3} lie on a triangle.

Then r_{1}, r_{2} and r_{3} are lineary dependent. We adopt the following rules: V will be a non-trivial real linear space

and o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} will be elements of the points of the projective space over V. The following proposition is true

(7) Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear. In the sequel u, v, w, y will denote vectors of V. A projective space defined

in terms of collinearity is said to be a Desarguesian projective space defined in terms of collinearity if: (Def.4) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state three propositions:

(8) Let C_{1} be a projective space defined in terms of collinearity. Then C_{1} is a Desarguesian projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(9) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then the projective space over V is a Desarguesian projective space defined in terms of collinearity.

(10) Let C_{1} be a collinearity structure. Then C_{1} is a Desarguesian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(iv) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of C_{1} such that p, q and r are not collinear,

(vi) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

A Fanoian projective space defined in terms of collinearity is called a Fano-Desarguesian projective space defined in terms of collinearity if: (Def.5) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear, (x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

One can prove the following propositions:

(11) Let C_{1} be a Fanoian projective space defined in terms of collinearity. Then C_{1} is a Fano-Desarguesian projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(12) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then the projective space over V is a Fano-Desarguesian projective space defined in terms of collinearity.

(13) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(iv) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of C_{1} such that p, q and r are not collinear,

(vi) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(14) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Desarguesian projective space defined in terms of collinearity, (ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1},

r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

A projective plane defined in terms of collinearity is called a Desarguesian projective plane defined in terms of collinearity if: (Def.6) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state four propositions:

(15) Let C_{1} be a projective plane defined in terms of collinearity. Then C_{1} is a Desarguesian projective plane defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(16) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then the projective space over V is a Desarguesian projective plane defined in terms of collinearity.

(17) Let C_{1} be a collinearity structure. Then C_{1} is a Desarguesian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) there exist elements p, q, r of the points of C_{1} such that p, q and r are

not collinear,

(v) for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(18) For every C_{1} being a collinearity structure holds C_{1} is a Desarguesian projective plane defined in terms of collinearity if and only if C_{1} is a Desarguesian projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A Fanoian projective plane defined in terms of collinearity is called a Fano-Desarguesian projective plane defined in terms of collinearity if: (Def.7) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

One can prove the following propositions:

(19) Let C_{1} be a Fanoian projective plane defined in terms of collinearity. Then C_{1} is a Fano-Desarguesian projective plane defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(20) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then the projective space over V is a Fano-Desarguesian projective plane defined in terms of collinearity.

(21) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) there exist elements p, q, r of the points of C_{1} such that p, q and r are

not collinear,

(v) for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(22) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Desarguesian projective plane defined in terms of collinearity, (ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1},

r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(23) For every C_{1} being a collinearity structure holds C_{1} is a Fano-Desarguesian projective plane defined in terms of collinearity if and only if C_{1} is a Fano-Desarguesian projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.

[3] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part I. Formalized Mathematics, 1(4):767–776, 1990.

[4] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.

Projective Spaces - part IV

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. A continuation of [4]. In the classes of projective spaces, defined in terms of collinearity, introduced in the article [3], we distinguish the subclasses of Desarguesian projective structures. As examples of these objects we consider analytical projective spaces defined over suitable real linear spaces.

MML Identifier: ANPROJ 5.

The notation and terminology used here have been introduced in the following papers: [1], [5], [2], [3], and [4]. We adopt the following convention: a, b, c, d denote real numbers, V denotes a non-trivial real linear space, and u, v, w, y, u_{1} denote vectors of V. An at least 3 dimensional projective space defined in terms of collinearity is said to be a Desarguesian at least 3 dimensional projective space defined in terms of collinearity if: (Def.1) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3},

(vi) p_{3} ≠ q_{3},

(vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

Then r_{1}, r_{2} and r_{3} are collinear.

The following propositions are true:

(1) Let C_{1} be an at least 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(2) If there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0, then the projective space over V is a Desarguesian at least 3 dimensional projective space defined in terms of collinearity.

(3) Let C_{1} be a collinearity structure. Then C_{1} is a Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(4) For every C_{1} being a collinearity structure holds C_{1} is a Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Desarguesian projective space defined in terms of collinearity and there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A Fanoian at least 3 dimensional projective space defined in terms of collinearity is called a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity if: (Def.2) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state several propositions:

(5) Let C_{1} be a Fanoian at least 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(6) If there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0, then the projective space over V is a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity.

(7) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vi) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(8) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Desarguesian at least 3 dimensional projective space defined in terms of collinearity,

(ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(9) For every C_{1} being a collinearity structure holds C_{1} is a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Fano-Desarguesian projective space defined in terms of collinearity and there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A 3 dimensional projective space defined in terms of collinearity is called a Desarguesian 3 dimensional projective space defined in terms of collinearity if: (Def.3) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state four propositions:

(10) Let C_{1} be a 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(11) Suppose that

(i) there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}. Then the projective space over V is a Desarguesian 3 dimensional projective space defined in terms of collinearity.

(12) Let C_{1} be a collinearity structure. Then C_{1} is a Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(13) For every C_{1} being a collinearity structure holds C_{1} is a Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Desarguesian at least 3 dimensional projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

A Fanoian 3 dimensional projective space defined in terms of collinearity is called a Fano-Desarguesian 3 dimensional projective space defined in terms of collinearity if: (Def.4) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ q_{1}, (ii) p_{1} ≠ q_{1}, (iii) o ≠ q_{2}, (iv) p_{2} ≠ q_{2},

(v) o ≠ q_{3}, (vi) p_{3} ≠ q_{3}, (vii) o, p_{1} and p_{2} are not collinear, (viii) o, p_{1} and p_{3} are not collinear, (ix) o, p_{2} and p_{3} are not collinear,

(x) p_{1}, p_{2} and r_{3} are collinear, (xi) q_{1}, q_{2} and r_{3} are collinear, (xii) p_{2}, p_{3} and r_{1} are collinear, (xiii) q_{2}, q_{3} and r_{1} are collinear, (xiv) p_{1}, p_{3} and r_{2} are collinear, (xv) q_{1}, q_{3} and r_{2} are collinear, (xvi) o, p_{1} and q_{1} are collinear, (xvii) o, p_{2} and q_{2} are collinear, (xviii) o, p_{3} and q_{3} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state several propositions:

(14) Let C_{1} be a Fanoian 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Fano-Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(15) Suppose that

(i) there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}. Then the projective space over V is a Fano-Desarguesian 3 dimensional projective space defined in terms of collinearity.

(16) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vi) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vii) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear,

(viii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ q_{1} and p_{1} ≠ q_{1} and o ≠ q_{2} and p_{2} ≠ q_{2} and o ≠ q_{3} and p_{3} ≠ q_{3} and o, p_{1} and p_{2} are not collinear and o, p_{1} and p_{3} are not collinear and o, p_{2} and p_{3} are not collinear and p_{1}, p_{2} and r_{3} are collinear and q_{1}, q_{2} and r_{3} are collinear and p_{2}, p_{3} and r_{1} are collinear and q_{2}, q_{3} and r_{1} are collinear and p_{1}, p_{3} and r_{2} are collinear and q_{1}, q_{3} and r_{2} are collinear and o, p_{1} and q_{1} are collinear and o, p_{2} and q_{2} are collinear and o, p_{3} and q_{3} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(17) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Desarguesian 3 dimensional projective space defined in terms of collinearity,

(ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(18) For every C_{1} being a collinearity structure holds C_{1} is a Fano-Desarguesian 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Fano-Desarguesian at least 3 dimensional projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.

[3] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part II. Formalized Mathematics, 1(5):901–907, 1990.

[4] Wojciech Leoṅczuk and Krzysztof Prażmowski. Projective spaces - part III. Formalized Mathematics, 1(5):909–918, 1990.

[5] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.

928

Projective Spaces - part V

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. In the classes of projective spaces, defined in terms of collinearity, introduced in the article [3], we distinguish the subclasses of Pappian projective structures. As examples of these objects we consider analytical projective spaces defined over suitable real linear spaces; analytical counterpart of the Pappus Axiom is proved without any assumption on the dimension of the underlying linear space.

MML Identifier: ANPROJ 6.

The terminology and notation used in this paper are introduced in the following papers: [1], [5], [2], [3], and [4]. We follow a convention: V will denote a real linear space, o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} will denote vectors of V, and a, b, c will denote real numbers. Let us consider V, o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}. We say that o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, and q_{3} lie on an angle if and only if: (Def.1) o, p_{1} and q_{1} are not lineary dependent and o, p_{1} and p_{2} are lineary

dependent and o, p_{1} and p_{3} are lineary dependent and o, q_{1} and q_{2} are lineary dependent and o, q_{1} and q_{3} are lineary dependent.

One can prove the following proposition

(1) o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, and q_{3} lie on an angle if and only if o, p_{1} and q_{1} are not lineary dependent and o, p_{1} and p_{2} are lineary dependent and o, p_{1} and p_{3} are lineary dependent and o, q_{1} and q_{2} are lineary dependent and o, q_{1} and q_{3} are lineary dependent.

Let us consider V, o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}. We say that o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3} are half-mutually not proportional if and only if: (Def.2) o and p_{2} are not proportional and o and p_{3} are not proportional and o

and q_{2} are not proportional and o and q_{3} are not proportional and p_{1} and p_{2} are not proportional and p_{1} and p_{3} are not proportional and q_{1} and q_{2} are not proportional and q_{1} and q_{3} are not proportional and p_{2} and p_{3} are not proportional and q_{2} and q_{3} are not proportional.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

Next we state two propositions:

(2) o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3} are half-mutually not proportional if and only if the following conditions are satisfied:

(i) o and p_{2} are not proportional, (ii) o and p_{3} are not proportional, (iii) o and q_{2} are not proportional, (iv) o and q_{3} are not proportional,

(v) p_{1} and p_{2} are not proportional, (vi) p_{1} and p_{3} are not proportional, (vii) q_{1} and q_{2} are not proportional, (viii) q_{1} and q_{3} are not proportional, (ix) p_{2} and p_{3} are not proportional,

(x) q_{2} and q_{3} are not proportional.

(3) Suppose that

(i) o is a proper vector, (ii) p_{1}, p_{2} and p_{3} are proper vectors, (iii) q_{1}, q_{2} and q_{3} are proper vectors, (iv) r_{1}, r_{2} and r_{3} are proper vectors,

(v) o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, and q_{3} lie on an angle, (vi) o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3} are half-mutually not proportional, (vii) p_{1}, q_{2} and r_{3} are lineary dependent, (viii) q_{1}, p_{2} and r_{3} are lineary dependent, (ix) p_{1}, q_{3} and r_{2} are lineary dependent,

(x) p_{3}, q_{1} and r_{2} are lineary dependent, (xi) p_{2}, q_{3} and r_{1} are lineary dependent, (xii) p_{3}, q_{2} and r_{1} are lineary dependent.

Then r_{1}, r_{2} and r_{3} are lineary dependent. We adopt the following convention: V will denote a non-trivial real linear

space and o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} will denote elements of the points of the projective space over V. The following proposition is true

(4) Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear. In the sequel u, v, w, y are vectors of V. A projective space defined in

terms of collinearity is said to be a Pappian projective space defined in terms of collinearity if: (Def.3) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state three propositions:

(5) Let C_{1} be a projective space defined in terms of collinearity. Then C_{1} is a Pappian projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(6) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then the projective space over V is a Pappian projective space defined in terms of collinearity.

(7) Let C_{1} be a collinearity structure. Then C_{1} is a Pappian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(iv) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of C_{1} such that p, q and r are not collinear,

(vi) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

A Fanoian projective space defined in terms of collinearity is said to be a Fano-Pappian projective space defined in terms of collinearity if: (Def.4) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state four propositions:

(8) Let C_{1} be a Fanoian projective space defined in terms of collinearity. Then C_{1} is a Fano-Pappian projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(9) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then the projective space over V is a Fano-Pappian projective space defined in terms of collinearity.

(10) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(iv) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of C_{1} such that p, q and r are not collinear, (vi) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1},

r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(11) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Pappian projective space defined in terms of collinearity, (ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1},

r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

A projective plane defined in terms of collinearity is called a Pappian projective plane defined in terms of collinearity if: (Def.5) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state four propositions:

(12) Let C_{1} be a projective plane defined in terms of collinearity. Then C_{1} is a Pappian projective plane defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(13) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then the projective space over V is a Pappian projective plane defined in terms of collinearity.

(14) Let C_{1} be a collinearity structure. Then C_{1} is a Pappian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) there exist elements p, q, r of the points of C_{1} such that p, q and r are

not collinear,

(v) for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(15) For every C_{1} being a collinearity structure holds C_{1} is a Pappian projective plane defined in terms of collinearity if and only if C_{1} is a Pappian projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A Fanoian projective plane defined in terms of collinearity is called a Fano-Pappian projective plane defined in terms of collinearity if: (Def.6) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state several propositions:

(16) Let C_{1} be a Fanoian projective plane defined in terms of collinearity. Then C_{1} is a Fano-Pappian projective plane defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(17) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then the projective space over V is a Fano-Pappian projective plane defined in terms of collinearity.

(18) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) there exist elements p, q, r of the points of C_{1} such that p, q and r are

not collinear,

(v) for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(19) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Pappian projective plane defined in terms of collinearity, (ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1},

r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or

r_{2}, r_{1} and q_{1} are collinear.

(20) For every C_{1} being a collinearity structure holds C_{1} is a Fano-Pappian projective plane defined in terms of collinearity if and only if C_{1} is a Fano-Pappian projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1} of the points of C_{1} there exists an element r of the points of C_{1} such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.

[3] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part I. Formalized Mathematics, 1(4):767–776, 1990.

[4] Wojciech Leoṅczuk and Krzysztof Prażmowski. Projective spaces - part III. Formalized Mathematics, 1(5):909–918, 1990.

[5] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.

Projective Spaces - part VI

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. The article is a continuation of [4]. In the classes of projective spaces, defined in terms of collinearity, introduced in the article [3], we distinguish the subclasses of Pappian projective structures. As examples of these types of objects we consider analytical projective spaces defined over suitable real linear spaces.

MML Identifier: ANPROJ 7.

The terminology and notation used in this paper have been introduced in the following articles: [1], [5], [2], [3], and [4]. We adopt the following rules: a, b, c, d will be real numbers, V will be a non-trivial real linear space, and u, v, w, y, u_{1} will be vectors of V. An at least 3 dimensional projective space defined in terms of collinearity is said to be a Pappian at least 3 dimensional projective space defined in terms of collinearity if: (Def.1) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

Then r_{1}, r_{2} and r_{3} are collinear.

We now state four propositions:

(1) Let C_{1} be an at least 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(2) If there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0, then the projective space over V is a Pappian at least 3 dimensional projective space defined in terms of collinearity.

(3) Let C_{1} be a collinearity structure. Then C_{1} is a Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear, (vi) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1}

such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(4) For every C_{1} being a collinearity structure holds C_{1} is a Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Pappian projective space defined in terms of collinearity and there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A Fanoian at least 3 dimensional projective space defined in terms of collinearity is called a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity if: (Def.2) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

One can prove the following propositions:

(5) Let C_{1} be a Fanoian at least 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(6) If there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0, then the projective space over V is a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity.

(7) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vi) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(8) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) C_{1} is a Pappian at least 3 dimensional projective space defined in terms of collinearity,

(ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(9) For every C_{1} being a collinearity structure holds C_{1} is a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Fano-Pappian projective space defined in terms of collinearity and there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A 3 dimensional projective space defined in terms of collinearity is called a Pappian 3 dimensional projective space defined in terms of collinearity if: (Def.3) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

The following four propositions are true:

(10) Let C_{1} be a 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Pappian 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(11) Suppose that

(i) there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}. Then the projective space over V is a Pappian 3 dimensional projective space defined in terms of collinearity.

(12) Let C_{1} be a collinearity structure. Then C_{1} is a Pappian 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear,

(vii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear

and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(13) For every C_{1} being a collinearity structure holds C_{1} is a Pappian 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Pappian at least 3 dimensional projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

A Fanoian 3 dimensional projective space defined in terms of collinearity is called a Fano-Pappian 3 dimensional projective space defined in terms of collinearity if: (Def.4) Let o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} be elements of the points of it .

Suppose that

(i) o ≠ p_{2}, (ii) o ≠ p_{3}, (iii) p_{2} ≠ p_{3}, (iv) p_{1} ≠ p_{2},

(v) p_{1} ≠ p_{3}, (vi) o ≠ q_{2}, (vii) o ≠ q_{3}, (viii) q_{2} ≠ q_{3}, (ix) q_{1} ≠ q_{2},

(x) q_{1} ≠ q_{3}, (xi) o, p_{1} and q_{1} are not collinear, (xii) o, p_{1} and p_{2} are collinear, (xiii) o, p_{1} and p_{3} are collinear, (xiv) o, q_{1} and q_{2} are collinear, (xv) o, q_{1} and q_{3} are collinear, (xvi) p_{1}, q_{2} and r_{3} are collinear, (xvii) q_{1}, p_{2} and r_{3} are collinear, (xviii) p_{1}, q_{3} and r_{2} are collinear, (xix) p_{3}, q_{1} and r_{2} are collinear, (xx) p_{2}, q_{3} and r_{1} are collinear, (xxi) p_{3}, q_{2} and r_{1} are collinear.

Then r_{1}, r_{2} and r_{3} are collinear.

The following propositions are true:

(14) Let C_{1} be a Fanoian 3 dimensional projective space defined in terms of collinearity. Then C_{1} is a Fano-Pappian 3 dimensional projective space defined in terms of collinearity if and only if for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(15) Suppose that

(i) there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}. Then the projective space over V is a Fano-Pappian 3 dimensional projective space defined in terms of collinearity.

(16) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian 3 dimensional projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r of the points of C_{1} holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear,

(ii) for all elements p, q, r, r_{1}, r_{2} of the points of C_{1} such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(iii) for every elements p, q of the points of C_{1} there exists an element r of

the points of C_{1} such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of C_{1} such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of C_{1} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(v) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear,

(vi) there exist elements p, p_{1}, q, q_{1} of the points of C_{1} such that for no element r of the points of C_{1} holds p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vii) for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear,

(viii) for all elements o, p_{1}, p_{2}, p_{3}, q_{1}, q_{2}, q_{3}, r_{1}, r_{2}, r_{3} of the points of C_{1} such that o ≠ p_{2} and o ≠ p_{3} and p_{2} ≠ p_{3} and p_{1} ≠ p_{2} and p_{1} ≠ p_{3} and o ≠ q_{2} and o ≠ q_{3} and q_{2} ≠ q_{3} and q_{1} ≠ q_{2} and q_{1} ≠ q_{3} and o, p_{1} and q_{1} are not collinear and o, p_{1} and p_{2} are collinear and o, p_{1} and p_{3} are collinear and o, q_{1} and q_{2} are collinear and o, q_{1} and q_{3} are collinear and p_{1}, q_{2} and r_{3} are collinear and q_{1}, p_{2} and r_{3} are collinear and p_{1}, q_{3} and r_{2} are collinear and p_{3}, q_{1} and r_{2} are collinear and p_{2}, q_{3} and r_{1} are collinear and p_{3}, q_{2} and r_{1} are collinear holds r_{1}, r_{2} and r_{3} are collinear.

(17) Let C_{1} be a collinearity structure. Then C_{1} is a Fano-Pappian 3 dimensional projective space defined in terms of collinearity if and only if the

following conditions are satisfied:

(i) C_{1} is a Pappian 3 dimensional projective space defined in terms of collinearity,

(ii) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of C_{1} such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(18) For every C_{1} being a collinearity structure holds C_{1} is a Fano-Pappian 3 dimensional projective space defined in terms of collinearity if and only if C_{1} is a Fano-Pappian at least 3 dimensional projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1}, r_{2} of the points of C_{1} there exist elements r, r_{1} of the points of C_{1} such that p, q and r are collinear and p_{1}, q_{1} and r_{1} are collinear and r_{2}, r and r_{1} are collinear.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.

[3] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part II. Formalized Mathematics, 1(5):901–907, 1990.

[4] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part V. Formalized Mathematics, 1(5):929–938, 1990.

[5] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.

948

Some Elementary Notions of the Theory of Petri Nets

Waldemar Korczyński^{1} Warsaw University Białystok

Summary. Some fundamental notions of the theory of Petri nets are described in Mizar formalism. A Petri net is defined as a triple of the form ⟨places, transitions, flow⟩ with places and transitions being disjoint sets and flow being a relation included in places × transitions.

MML Identifier: NET 1.

The notation and terminology used here have been introduced in the following articles: [1], and [2]. In the sequel x, y will be arbitrary. We consider nets which are systems

⟨places, transitions, a flow relation⟩, where the places constitute a set, the transitions constitute a set, and the flow relation is a binary relation. In the sequel N is a net. Let N be a net. We say that N is a Petri net if and only if: (Def.1) (the places of N)∩ (the transitions of N) = ∅ and the flow relation of

N ⊆ [: the places of N, the transitions of N :] ∪ [: the transitions of N, the places of N :].

Let N be a net. The functor Elements(N) yielding a set is defined as follows: (Def.2) Elements(N) = (the places of N)∪ (the transitions of N).

We now state several propositions:

(1) For every N and for every x such that Elements(N) ≠ ∅ holds x is an element of Elements(N) if and only if x ∈ Elements(N).

(2) For every N and for every x such that the places of N ≠ ∅ holds x is an element of the places of N if and only if x ∈ the places of N.

1Supported by RPBP.III-24.C1.

(3) For every N and for every x such that the transitions of N ≠ ∅ holds x is an element of the transitions of N if and only if x ∈ the transitions of N.

(4) For every N holds the places of N ⊆ Elements(N).

(5) For every N holds the transitions of N ⊆ Elements(N). Let N be a net. A set is said to be an element of N if:

(Def.3) it = Elements(N).

Next we state several propositions:

(6) For every N and for every x holds x ∈ Elements(N) if and only if x ∈ the places of N or x ∈ the transitions of N.

(7) For every N and for every x such that Elements(N) ≠ ∅ holds if x is an element of Elements(N), then x is an element of the places of N or x is an element of the transitions of N.

(8) For every N and for every x such that x is an element of the places of N and the places of N ≠ ∅ holds x is an element of Elements(N).

(9) For every N and for every x such that x is an element of the transitions of N and the transitions of N ≠ ∅ holds x is an element of Elements(N).

(10) ⟨∅, ∅, ⟩ is a Petri net.

A net is said to be a Petri net if: (Def.4) it is a Petri net.

We now state several propositions:

(11) For every Petri net N holds it is not true that: x ∈ the places of N and x ∈ the transitions of N.

(12) For every Petri net N and for all x, y such that ⟨⟨x, y⟩⟩ ∈ the flow relation of N and x ∈ the transitions of N holds y ∈ the places of N.

(13) For every Petri net N and for all x, y such that ⟨⟨x, y⟩⟩ ∈ the flow relation of N and y ∈ the transitions of N holds x ∈ the places of N.

(14) For every Petri net N and for all x, y such that ⟨⟨x, y⟩⟩ ∈ the flow relation of N and x ∈ the places of N holds y ∈ the transitions of N.

(15) For every Petri net N and for all x, y such that ⟨⟨x, y⟩⟩ ∈ the flow relation

of N and y ∈ the places of N holds x ∈ the transitions of N. We now define two new predicates. Let N be a Petri net, and let us consider

x, y. We say that x is a pre-element of y in N if and only if: (Def.5) ⟨⟨y, x⟩⟩ ∈ the flow relation of N and x ∈ the transitions of N. We say that x is a post-element of y in N if and only if: (Def.6) ⟨⟨x, y⟩⟩ ∈ the flow relation of N and x ∈ the transitions of N.

We now define two new functors. Let N be a net, and let x be an element of Elements(N). The functor Pre(N, x) yielding a set is defined by: (Def.7) y ∈ Pre(N, x) if and only if y ∈ Elements(N) and ⟨⟨y, x⟩⟩ ∈ the flow

relation of N.

The functor Post(N, x) yielding a set is defined by: (Def.8) y ∈ Post(N, x) if and only if y ∈ Elements(N) and ⟨⟨x, y⟩⟩ ∈ the flow

relation of N.

Next we state several propositions:

(16) For every Petri net N and for every element x of Elements(N) holds Pre(N, x) ⊆ Elements(N).

(17) For every Petri net N and for every element x of Elements(N) holds Pre(N, x) ∈ 2^{Elements(N)}.

(18) For every Petri net N and for every element x of Elements(N) holds Post(N, x) ⊆ Elements(N).

(19) For every Petri net N and for every element x of Elements(N) holds Post(N, x) ∈ 2^{Elements(N)}.

(20) For every Petri net N and for every element y of Elements(N) such that y ∈ the transitions of N holds x ∈ Pre(N, y) if and only if y is a pre-element of x in N.

(21) For every Petri net N and for every element y of Elements(N) such that y ∈ the transitions of N holds x ∈ Post(N, y) if and only if y is a post-element of x in N.

Let N be a Petri net, and let x be an element of Elements(N). Let us assume that Elements(N) ≠ ∅. The functor enter(N, x) yielding a set is defined by: (Def.9) if x ∈ the places of N, then enter(N, x) = {x} but if x ∈ the transitions

of N, then enter(N, x) = Pre(N, x).

We now state three propositions:

(22) For every Petri net N and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds enter(N, x) = {x} or enter(N, x) = Pre(N, x).

(23) For every Petri net N and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds enter(N, x) ⊆ Elements(N).

(24) For every Petri net N and for every element x of Elements(N) such

that Elements(N) ≠ ∅ holds enter(N, x) ∈ 2^{Elements(N)}. Let N be a Petri net, and let x be an element of Elements(N). Let us assume

that Elements(N) ≠ ∅. The functor exit(N, x) yields a set and is defined by: (Def.10) if x ∈ the places of N, then exit(N, x) = {x} but if x ∈ the transitions

of N, then exit(N, x) = Post(N, x).

We now state three propositions:

(25) For every Petri net N and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds exit(N, x) = {x} or exit(N, x) = Post(N, x).

(26) For every Petri net N and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds exit(N, x) ⊆ Elements(N).

(27) For every Petri net N and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds exit(N, x) ∈ 2^{Elements(N)}.

Let N be a Petri net, and let x be an element of Elements(N). Let us assume that Elements(N) ≠ ∅. The functor field(N, x) yielding a set is defined as follows:

(Def.11) field(N, x) = enter(N, x) ∪ exit(N, x).

We now define two new functors. Let N be a net, and let x be an element of the transitions of N. The functor Prec(N, x) yielding a set is defined by:

(Def.12) y ∈ Prec(N, x) if and only if y ∈ the places of N and ⟨⟨y, x⟩⟩ ∈ the flow

relation of N. The functor Postc(N, x) yielding a set is defined as follows:

(Def.13) y ∈ Postc(N, x) if and only if y ∈ the places of N and ⟨⟨x, y⟩⟩ ∈ the flow relation of N.

We now define two new functors. Let N be a Petri net, and let X be a set. Let us assume that X ⊆ Elements(N). The functor Entr(N, X) yields a set and is defined by:

(Def.14) x ∈ Entr(N, X) if and only if x ∈ 2^{Elements(N)} and there exists an

element y of Elements(N) such that y ∈ X and x = enter(N, y). The functor Ext(N, X) yielding a set is defined by:

(Def.15) x ∈ Ext(N, X) if and only if x ∈ 2^{Elements(N)} and there exists an element y of Elements(N) such that y ∈ X and x = exit(N, y).

Next we state two propositions:

(28) For every Petri net N and for every element x of Elements(N) and for every set X such that Elements(N) ≠ ∅ and X ⊆ Elements(N) and x ∈ X holds enter(N, x) ∈ Entr(N, X).

(29) For every Petri net N and for every element x of Elements(N) and for every set X such that Elements(N) ≠ ∅ and X ⊆ Elements(N) and x ∈ X holds exit(N, x) ∈ Ext(N, X).

We now define two new functors. Let N be a Petri net, and let X be a set. Let us assume that X ⊆ Elements(N). The functor Input(N, X) yields a set and is defined by:

(Def.16) Input(N, X) = ⋃ Entr(N, X).

The functor Output(N, X) yielding a set is defined by: (Def.17) Output(N, X) = ⋃ Ext(N, X).

The following four propositions are true:

(30) For every Petri net N and for every x and for every set X such that Elements(N) ≠ ∅ and X ⊆ Elements(N) holds x ∈ Input(N, X) if and only if there exists an element y of Elements(N) such that y ∈ X and x ∈ enter(N, y).

(31) For every Petri net N and for every x and for every set X such that Elements(N) ≠ ∅ and X ⊆ Elements(N) holds x ∈ Output(N, X) if and only if there exists an element y of Elements(N) such that y ∈ X and x ∈ exit(N, y).

(32) Let N be a Petri net. Then for every subset X of Elements(N) and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds x ∈ Input(N, X) if and only if x ∈ X and x ∈ the places of N or there exists an element y of Elements(N) such that y ∈ X and y ∈ the transitions of N and y is a pre-element of x in N.

(33) Let N be a Petri net. Then for every subset X of Elements(N) and for every element x of Elements(N) such that Elements(N) ≠ ∅ holds x ∈ Output(N, X) if and only if x ∈ X and x ∈ the places of N or there exists an element y of Elements(N) such that y ∈ X and y ∈ the transitions of N and y is a post-element of x in N.

References

[1] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[2] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

954

Classes of Conjugation. Normal Subgroups

Wojciech A. Trybulec^{1} Warsaw University

Summary. Theorems that were not proved in [8] and in [9] are discussed. In the article we define notion of conjugation for elements, subsets and subgroups of a group. We define the classes of conjugation. Normal subgroups of a group and normalizator of a subset of a group or of a subgroup are introduced. We also define the set of all subgroups of a group. An auxiliary theorem that belongs rather to [1] is proved.

MML Identifier: GROUP 3.

The papers [3], [10], [5], [2], [8], [9], [6], [4], and [7] provide the notation and terminology for this paper. For simplicity we follow a convention: x, y are arbitrary, X denotes a set, G denotes a group, a, b, c, d, g, h denote elements of G, A, B, C, D denote subsets of G, H, H_{1}, H_{2}, H_{3} denote subgroups of G, n denotes a natural number, and i denotes an integer. Next we state a number of propositions:

(1) (a·b)·b^{−1} = a and (a·b^{−1})·b = a and (b^{−1} ·b)·a = a and (b·b^{−1})·a = a and a·(b·b^{−1}) = a and a·(b^{−1}·b) = a and b^{−1}·(b·a) = a and b·(b^{−1}·a) = a.

(2) G is an Abelian group if and only if the operation of G is commutative.

(3) {1}_{G} is an Abelian group.

(4) If A ⊆ B and C ⊆ D, then A · C ⊆ B · D.

(5) If A ⊆ B, then a · A ⊆ a · B and A · a ⊆ B · a.

(6) If H_{1} is a subgroup of H_{2}, then a · H_{1} ⊆ a · H_{2} and H_{1} · a ⊆ H_{2} · a.

(7) a · H = {a} · H.

(8) H · a = H · {a}.

(9) (a · A) · H = a · (A · H).

(10) (A · a) · H = A · (a · H).

(11) (a · H) · A = a · (H · A).

(12) (A · H) · a = A · (H · a).

1Supported by RPBP.III-24.C1

(13) (H · a) · A = H · (a · A).

(14) (H · A) · a = H · (A · a).

(15) (H_{1} · a) · H_{2} = H_{1} · (a · H_{2}).

Let us consider G. The functor SubGr G yielding a non-empty set is defined by: (Def.1) x ∈ SubGr G if and only if x is a subgroup of G.

In the sequel D denotes a non-empty set. Next we state four propositions:

(16) If for every x holds x ∈ D if and only if x is a subgroup of G, then D = SubGr G.

(17) x ∈ SubGr G if and only if x is a subgroup of G.

(18) G ∈ SubGr G.

(19) If G is finite, then SubGr G is finite.

Let us consider G, a, b. The functor a^{b} yielding an element of G is defined as follows: (Def.2) a^{b} = (b^{−1} · a) · b.

One can prove the following propositions:

(20) a^{b} = (b^{−1} · a) · b and a^{b} = b^{−1} · (a · b).

(21) If a^{g} = b^{g}, then a = b.

(22) (1_{G})^{a} = 1_{G}.

(23) If a^{b} = 1_{G}, then a = 1_{G}.

(24) a^{1_{G}} = a.

(25) a^{a} = a.

(26) (a^{a})^{−1} = a and (a^{−1})^{a} = a^{−1}.

(27) a^{b} = a if and only if a · b = b · a.

(28) (a · b)^{g} = a^{g} · b^{g}.

(29) (a^{g})^{h} = a^{g·h}.

(30) ((a^{b})^{b})^{−1} = a and ((a^{b})^{−1})^{b} = a.

(31) a^{b} = c if and only if a = (c^{b})^{−1}.

(32) (a^{−1})^{b} = (a^{b})^{−1}.

(33) (a^{n})^{b} = (a^{b})^{n}.

(34) (a^{i})^{b} = (a^{b})^{i}.

(35) If G is an Abelian group, then a^{b} = a.

(36) If for all a, b holds a^{b} = a, then G is an Abelian group.

Let us consider G, A, B. The functor A^{B} yielding a subset of G is defined as follows: (Def.3) A^{B} = {g^{h} : g ∈ A ∧ h ∈ B}.

We now state a number of propositions:

(37) A^{B} = {g^{h} : g ∈ A ∧ h ∈ B}.

(38) x ∈ A^{B} if and only if there exist g, h such that x = g^{h} and g ∈ A and h ∈ B.

(39) A^{B} ≠ ∅ if and only if A ≠ ∅ and B ≠ ∅.

(40) A^{B} ⊆ (B^{−1} · A) · B.

(41) (A · B)^{C} ⊆ A^{C} · B^{C}.

(42) (A^{B})^{C} = A^{B·C}.

(43) (A^{−1})^{B} = (A^{B})^{−1}.

(44) {a}^{{b}} = {a^{b}}.

(45) {a}^{{b,c}} = {a^{b}, a^{c}}.

(46) {a, b}^{{c}} = {a^{c}, b^{c}}.

(47) {a, b}^{{c,d}} = {a^{c}, a^{d}, b^{c}, b^{d}}.

We now define two new functors. Let us consider G, A, g. The functor A^{g} yields a subset of G and is defined as follows: (Def.4) A^{g} = A^{{g}}. The functor g^{A} yields a subset of G and is defined by: (Def.5) g^{A} = {g}^{A}.

Next we state a number of propositions:

(48) A^{g} = A^{{g}}.

(49) g^{A} = {g}^{A}.

(50) x ∈ A^{g} if and only if there exists h such that x = h^{g} and h ∈ A.

(51) x ∈ g^{A} if and only if there exists h such that x = g^{h} and h ∈ A.

(52) g^{A} ⊆ (A^{−1} · g) · A.

(53) (A^{B})^{g} = A^{B·g}.

(54) (A^{g})^{B} = A^{g·B}.

(55) (g^{A})^{B} = g^{A·B}.

(56) (A^{a})^{b} = A^{a·b}.

(57) (a^{A})^{b} = a^{A·b}.

(58) (a^{b})^{A} = a^{b·A}.

(59) A^{g} = (g^{−1} · A) · g.

(60) (A · B)^{a} ⊆ A^{a} · B^{a}.

(61) A^{1_{G}} = A.

(62) If A ≠ ∅, then (1_{G})^{A} = {1_{G}}.

(63) ((A^{a})^{a})^{−1} = A and ((A^{a})^{−1})^{a} = A.

(64) A = B^{g} if and only if B = (A^{g})^{−1}.

(65) G is an Abelian group if and only if for all A, B such that B ≠ ∅ holds A^{B} = A.

(66) G is an Abelian group if and only if for all A, g holds A^{g} = A.

(67) G is an Abelian group if and only if for all A, g such that A ≠ ∅ holds g^{A} = {g}.

Let us consider G, H, a. The functor H^{a} yielding a subgroup of G is defined by: (Def.6) the carrier of H^{a} = \overline{H}^{a}.

The following propositions are true:

(68) If the carrier of H_{1} = \overline{H}^{a}, then H_{1} = H^{a}.

(69) The carrier of H^{a} = \overline{H}^{a}.

(70) x ∈ H^{a} if and only if there exists g such that x = g^{a} and g ∈ H.

(71) The carrier of H^{a} = (a^{−1} · H) · a.

(72) (H^{a})^{b} = H^{a·b}.

(73) H^{1_{G}} = H.

(74) ((H^{a})^{a})^{−1} = H and ((H^{a})^{−1})^{a} = H.

(75) H_{1} = H_{2}^{a} if and only if H_{2} = (H_{1}^{a})^{−1}.

(76) (H_{1} ∩ H_{2})^{a} = H_{1}^{a} ∩ H_{2}^{a}.

(77) Ord(H) = Ord(H^{a}).

(78) H is finite if and only if H^{a} is finite.

(79) If H is finite, then ord(H) = ord(H^{a}).

(80) {1}_{G}^{a} = {1}_{G}.

(81) If H^{a} = {1}_{G}, then H = {1}_{G}.

(82) Ω_{G}^{a} = G.

(83) If H^{a} = G, then H = G.

(84) |• : H| = |• : H^{a}|.

(85) If the left cosets of H is finite, then |• : H|_{}= |• : H^{a}|_{}.

(86) If G is an Abelian group, then for all H, a holds H^{a} = H.

Let us consider G, a, b. We say that a and b are conjugated if and only if: (Def.7) there exists g such that a = b^{g}.

We now state several propositions:

(87) a and b are conjugated if and only if there exists g such that a = b^{g}.

(88) a and b are conjugated if and only if there exists g such that b = a^{g}.

(89) a and a are conjugated.

(90) If a and b are conjugated, then b and a are conjugated.

(91) If a and b are conjugated and b and c are conjugated, then a and c are conjugated.

(92) If a and 1_{G} are conjugated or 1_{G} and a are conjugated, then a = 1_{G}.

(93) a\overline{Ω_{G}} = \{b : a and b are conjugated \}.

Let us consider G, a. The functor a^{•} yielding a subset of G is defined by: (Def.8) a^{•} = a\overline{Ω_{G}}.

We now state several propositions:

(94) a^{•} = a\overline{Ω_{G}}.

(95) x ∈ a^{•} if and only if there exists b such that b = x and a and b are conjugated.

(96) a ∈ b^{•} if and only if a and b are conjugated.

(97) a^{g} ∈ a^{•}.

(98) a ∈ a^{•}.

(99) If a ∈ b^{•}, then b ∈ a^{•}.

(100) a^{•} = b^{•} if and only if a^{•} meets b^{•}.

(101) a^{•} = {1_{G}} if and only if a = 1_{G}.

(102) a^{•} · A = A · a^{•}.

Let us consider G, A, B. We say that A and B are conjugated if and only if: (Def.9) there exists g such that A = B^{g}.

We now state several propositions:

(103) A and B are conjugated if and only if there exists g such that A = B^{g}.

(104) A and B are conjugated if and only if there exists g such that B = A^{g}.

(105) A and A are conjugated.

(106) If A and B are conjugated, then B and A are conjugated.

(107) If A and B are conjugated and B and C are conjugated, then A and C are conjugated.

(108) {a} and {b} are conjugated if and only if a and b are conjugated.

(109) If A and \overline{H_{1}} are conjugated, then there exists H_{2} such that the carrier

of H_{2} = A. Let us consider G, A. The functor A^{•} yielding a family of subsets of the

carrier of G is defined as follows: (Def.10) A^{•} = {B : A and B are conjugated }.

The following propositions are true:

(110) A^{•} = {B : A and B are conjugated }.

(111) x ∈ A^{•} if and only if there exists B such that x = B and A and B are conjugated.

(112) If x ∈ A^{•}, then x is a subset of G.

(113) A ∈ B^{•} if and only if A and B are conjugated.

(114) A^{g} ∈ A^{•}.

(115) A ∈ A^{•}.

(116) If A ∈ B^{•}, then B ∈ A^{•}.

(117) A^{•} = B^{•} if and only if A^{•} meets B^{•}.

(118) {a}^{•} = {{b} : b ∈ a^{•}}.

(119) If G is finite, then A^{•} is finite.

Let us consider G, H_{1}, H_{2}. We say that H_{1} and H_{2} are conjugated if and only if:

(Def.11) there exists g such that H_{1} = H_{2}^{g}.

The following propositions are true:

(120) H_{1} and H_{2} are conjugated if and only if there exists g such that H_{1} = H_{2}^{g}.

(121) H_{1} and H_{2} are conjugated if and only if there exists g such that H_{2} = H_{1}^{g}.

(122) H_{1} and H_{1} are conjugated.

(123) If H_{1} and H_{2} are conjugated, then H_{2} and H_{1} are conjugated.

(124) If H_{1} and H_{2} are conjugated and H_{2} and H_{3} are conjugated, then H_{1}

and H_{3} are conjugated. In the sequel L denotes a subset of SubGr G. Let us consider G, H. The

functor H^{•} yielding a subset of SubGr G is defined as follows: (Def.12) x ∈ H^{•} if and only if there exists H_{1} such that x = H_{1} and H and H_{1}

are conjugated. One can prove the following propositions:

(125) If for every x holds x ∈ L if and only if there exists H such that x = H and H_{1} and H are conjugated, then L = H_{1}^{•}.

(126) x ∈ H_{1}^{•} if and only if there exists H_{2} such that x = H_{2} and H_{1} and H_{2} are conjugated.

(127) If x ∈ H^{•}, then x is a subgroup of G.

(128) H_{1} ∈ H_{2}^{•} if and only if H_{1} and H_{2} are conjugated.

(129) H^{g} ∈ H^{•}.

(130) H ∈ H^{•}.

(131) If H_{1} ∈ H_{2}^{•}, then H_{2} ∈ H_{1}^{•}.

(132) H_{1}^{•} = H_{2}^{•} if and only if H_{1}^{•} meets H_{2}^{•}.

(133) If G is finite, then H^{•} is finite.

(134) H_{1} and H_{2} are conjugated if and only if \overline{H_{1}} and \overline{H_{2}} are conjugated.

Let us consider G. A subgroup of G is called a normal subgroup of G if: (Def.13) for every a holds it^{a} = it.

One can prove the following proposition

(135) If for every a holds H = H^{a}, then H is a normal subgroup of G. In the sequel N, N_{1}, N_{2} will denote ha normal subgroups of G. We now

state a number of propositions:

(136) N^{a} = N.

(137) {1}_{G} is a normal subgroup of G and Ω_{G} is a normal subgroup of G.

(138) N_{1} ∩ N_{2} is a normal subgroup of G.

(139) If G is an Abelian group, then H is a normal subgroup of G.

(140) H is a normal subgroup of G if and only if for every a holds a·H = H ·a.

(141) H is a normal subgroup of G if and only if for every a holds a·H ⊆ H ·a.

(142) H is a normal subgroup of G if and only if for every a holds H ·a ⊆ a·H.

(143) H is a normal subgroup of G if and only if for every A holds A·H = H·A.

(144) H is a normal subgroup of G if and only if for every a holds H is a subgroup of H^{a}.

(145) H is a normal subgroup of G if and only if for every a holds H^{a} is a subgroup of H.

(146) H is a normal subgroup of G if and only if H^{•} = {H}.

(147) H is a normal subgroup of G if and only if for every a such that a ∈ H holds a^{•} ⊆ \overline{H}.

(148) \overline{N_{1}} · \overline{N_{2}} = \overline{N_{2}} · \overline{N_{1}}.

(149) There exists N such that the carrier of N = \overline{N_{1}} · \overline{N_{2}}.

(150) The left cosets of N = the right cosets of N.

(151) If the left cosets of H is finite and |• : H|_{}= 2, then H is a normal

subgroup of G. Let us consider G, A. The functor N(A) yielding a subgroup of G is defined

by: (Def.14) the carrier of N(A) = {h : A^{h} = A}.

We now state several propositions:

(152) If the carrier of H = {h : A^{h} = A}, then H = N(A).

(153) The carrier of N(A) = {h : A^{h} = A}.

(154) x ∈ N(A) if and only if there exists h such that x = h and A^{h} = A.

(155) \overline{\overline{A^{•}}} = |• : N(A)|.

(156) If A^{•} is finite or the left cosets of N(A) is finite, then card A^{•} = |• : N(A)|_{}.

(157) \overline{\overline{a^{•}}} = |• : N(\{a\})|.

(158) If a^{•} is finite or the left cosets of N({a}) is finite, then card a^{•} = |• :

N({a})|_{}. Let us consider G, H. The functor N(H) yields a subgroup of G and is

defined as follows: (Def.15) N(H) = N(\overline{H}).

We now state several propositions:

(159) N(H) = N(\overline{H}).

(160) x ∈ N(H) if and only if there exists h such that x = h and H^{h} = H.

(161) \overline{\overline{H^{•}}} = |• : N(H)|.

(162) If H^{•} is finite or the left cosets of N(H) is finite, then card H^{•} = |• : N(H)|_{}.

(163) H is a normal subgroup of G if and only if N(H) = G.

(164) N({1}_{G}) = G.

(165) N(Ω_{G}) = G.

(166) If X is finite and card X = 2, then there exist x, y such that x ≠ y and X = {x, y}.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175– 180, 1990.

[4] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[5] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[6] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[7] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[8] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[9] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855–864, 1990.

[10] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

Replacing of Variables in Formulas of ZF Theory

Grzegorz Bancerek^{1} Warsaw University Białystok

Summary. Part one is a supplement to papers [1], [2], and [3]. It deals with concepts of selector functions, atomic, negative, conjunctive formulas and etc., subformulas, free variables, satisfiability and models (it is shown that axioms of the predicate and the quantifier calculus are satisfied in an arbitrary set). In part two there are introduced notions of variables occurring in a formula and replacing of variables in a formula.

MML Identifier: ZF LANG1.

The terminology and notation used in this paper have been introduced in the following articles: [9], [8], [5], [6], [4], [7], [1], and [2]. For simplicity we adopt the following rules: p, p_{1}, p_{2}, q, r, F, G, G_{1}, G_{2}, H, H_{1}, H_{2} will be ZF-formulae, x, x_{1}, x_{2}, y, y_{1}, y_{2}, z, z_{1}, z_{2}, s, t will be variables, a will be arbitrary, and X will be a set. Next we state a number of propositions:

(1) Var_{1}(x=y) = x and Var_{2}(x=y) = y.

(2) Var_{1}(xϵy) = x and Var_{2}(xϵy) = y.

(3) Arg(¬p) = p.

(4) LeftArg(p ∧ q) = p and RightArg(p ∧ q) = q.

(5) LeftArg(p ∨ q) = p and RightArg(p ∨ q) = q.

(6) Antecedent(p ⇒ q) = p and Consequent(p ⇒ q) = q.

(7) LeftSide(p ⇔ q) = p and RightSide(p ⇔ q) = q.

(8) Bound(∀_{x}p) = x and Scope(∀_{x}p) = p.

(9) Bound(∃_{x}p) = x and Scope(∃_{x}p) = p.

(10) p ∨ q = ¬p ⇒ q.

(11) If ∀_{x,y}p = ∀_{z}q, then x = z and ∀_{y}p = q.

1Supported by RPBP III-24.C1.

(12) If ∃_{x,y}p = ∃_{z}q, then x = z and ∃_{y}p = q.

(13) ∀_{x,y}p is universal and Bound(∀_{x,y}p) = x and Scope(∀_{x,y}p) = ∀_{y}p.

(14) ∃_{x,y}p is existential and Bound(∃_{x,y}p) = x and Scope(∃_{x,y}p) = ∃_{y}p.

(15) ∀_{x,y,z}p = ∀_{x}(∀_{y}(∀_{z}p)) and ∀_{x,y,z}p = ∀_{x,y}(∀_{z}p).

(16) If ∀_{x_{1},y_{1}}p_{1} = ∀_{x_{2},y_{2}}p_{2}, then x_{1} = x_{2} and y_{1} = y_{2} and p_{1} = p_{2}.

(17) If ∀_{x_{1},y_{1},z_{1}}p_{1} = ∀_{x_{2},y_{2},z_{2}}p_{2}, then x_{1} = x_{2} and y_{1} = y_{2} and z_{1} = z_{2} and p_{1} = p_{2}.

(18) If ∀_{x,y,z}p = ∀_{t}q, then x = t and ∀_{y,z}p = q.

(19) If ∀_{x,y,z}p = ∀_{t,s}q, then x = t and y = s and ∀_{z}p = q.

(20) If ∃_{x_{1},y_{1}}p_{1} = ∃_{x_{2},y_{2}}p_{2}, then x_{1} = x_{2} and y_{1} = y_{2} and p_{1} = p_{2}.

(21) ∃_{x,y,z}p = ∃_{x}(∃_{y}(∃_{z}p)) and ∃_{x,y,z}p = ∃_{x,y}(∃_{z}p).

(22) If ∃_{x_{1},y_{1},z_{1}}p_{1} = ∃_{x_{2},y_{2},z_{2}}p_{2}, then x_{1} = x_{2} and y_{1} = y_{2} and z_{1} = z_{2} and p_{1} = p_{2}.

(23) If ∃_{x,y,z}p = ∃_{t}q, then x = t and ∃_{y,z}p = q.

(24) If ∃_{x,y,z}p = ∃_{t,s}q, then x = t and y = s and ∃_{z}p = q.

(25) ∀_{x,y,z}p is universal and Bound(∀_{x,y,z}p) = x and Scope(∀_{x,y,z}p) = ∀_{y,z}p.

(26) ∃_{x,y,z}p is existential and Bound(∃_{x,y,z}p) = x and Scope(∃_{x,y,z}p) = ∃_{y,z}p.

(27) If H is disjunctive, then LeftArg(H) = Arg(LeftArg(Arg(H))).

(28) If H is disjunctive, then RightArg(H) = Arg(RightArg(Arg(H))).

(29) If H is conditional, then Antecedent(H) = LeftArg(Arg(H)).

(30) If H is conditional, then Consequent(H) = Arg(RightArg(Arg(H))).

(31) If H is biconditional, then LeftSide(H) = Antecedent(LeftArg(H)) and LeftSide(H) = Consequent(RightArg(H)).

(32) If H is biconditional, then RightSide(H) = Consequent(LeftArg(H)) and RightSide(H) = Antecedent(RightArg(H)).

(33) If H is existential, then Bound(H) = Bound(Arg(H)) and Scope(H) = Arg(Scope(Arg(H))).

(34) Arg(F ∨G) = ¬F ∧¬G and Antecedent(F ∨G) = ¬F and Consequent(F ∨ G) = G.

(35) Arg(F ⇒ G) = F ∧ ¬G.

(36) LeftArg(F ⇔ G) = F ⇒ G and RightArg(F ⇔ G) = G ⇒ F.

(37) Arg(∃_{x}H) = ∀_{x}¬H.

(38) If H is disjunctive, then H is conditional and H is negative and Arg(H) is conjunctive and LeftArg(Arg(H)) is negative and RightArg(Arg(H)) is negative.

(39) If H is conditional, then H is negative and Arg(H) is conjunctive and RightArg(Arg(H)) is negative.

(40) If H is biconditional, then H is conjunctive and LeftArg(H) is conditional and RightArg(H) is conditional.

(41) If H is existential, then H is negative and Arg(H) is universal and Scope(Arg(H)) is negative.

(42) It is not true that: H is an equality and H is a membership or H is negative or H is conjunctive or H is universal and it is not true that: H is a membership and H is negative or H is conjunctive or H is universal and it is not true that: H is negative and H is conjunctive or H is universal and it is not true that: H is conjunctive and H is universal.

(43) If F is a subformula of G, then len F ≤ len G.

(44) Suppose F is a proper subformula of G and G is a subformula of H or F is a subformula of G and G is a proper subformula of H or F is a subformula of G and G is an immediate constituent of H or F is an immediate constituent of G and G is a subformula of H or F is a proper subformula of G and G is an immediate constituent of H or F is an immediate constituent of G and G is a proper subformula of H. Then F is a proper subformula of H.

(45) H is not a proper subformula of H.

(46) H is not an immediate constituent of H.

(47) It is not true that: G is a proper subformula of H and H is a subformula of G.

(48) It is not true that: G is a proper subformula of H and H is a proper subformula of G.

(49) It is not true that: G is a subformula of H and H is an immediate constituent of G.

(50) It is not true that: G is a proper subformula of H and H is an immediate constituent of G.

(51) If ¬F is a subformula of H, then F is a proper subformula of H.

(52) If F ∧ G is a subformula of H, then F is a proper subformula of H and G is a proper subformula of H.

(53) If ∀_{x}H is a subformula of F, then H is a proper subformula of F.

(54) F ∧ ¬G is a proper subformula of F ⇒ G and F is a proper subformula of F ⇒ G and ¬G is a proper subformula of F ⇒ G and G is a proper subformula of F ⇒ G.

(55) ¬F ∧¬G is a proper subformula of F ∨G and ¬F is a proper subformula of F ∨ G and ¬G is a proper subformula of F ∨ G and F is a proper subformula of F ∨ G and G is a proper subformula of F ∨ G.

(56) ∀_{x}¬H is a proper subformula of ∃_{x}H and ¬H is a proper subformula of ∃_{x}H.

(57) G is a subformula of H if and only if G ∈ Subformulae H.

(58) If G ∈ Subformulae H, then Subformulae G ⊆ Subformulae H.

(59) H ∈ Subformulae H.

(60) Subformulae F ⇒ G = (Subformulae F ∪ Subformulae G) ∪ {¬G, F ∧ ¬G, F ⇒ G}.

(61) Subformulae F ∨G = (Subformulae F ∪Subformulae G)∪{¬G, ¬F, ¬F ∧ ¬G, F ∨ G}.

(62) Subformulae F ⇔ G = (Subformulae F ∪ Subformulae G) ∪ {¬G, F ∧ ¬G, F ⇒ G, ¬F, G ∧ ¬F, G ⇒ F, F ⇔ G}.

(63) Free(x=y) = {x, y}.

(64) Free(xϵy) = {x, y}.

(65) Free(¬p) = Free p.

(66) Free(p ∧ q) = Free p ∪ Free q.

(67) Free(∀_{x}p) = Free p \ {x}.

(68) Free(p ∨ q) = Free p ∪ Free q.

(69) Free(p ⇒ q) = Free p ∪ Free q.

(70) Free(p ⇔ q) = Free p ∪ Free q.

(71) Free(∃_{x}p) = Free p \ {x}.

(72) Free(∀_{x,y}p) = Free p \ {x, y}.

(73) Free(∀_{x,y,z}p) = Free p \ {x, y, z}.

(74) Free(∃_{x,y}p) = Free p \ {x, y}.

(75) Free(∃_{x,y,z}p) = Free p \ {x, y, z}.

The scheme ZF Induction deals with a unary predicate P, and states that: for every H holds P[H]

provided the parameter satisfies the following conditions:

• for all x_{1}, x_{2} holds P[x_{1}=x_{2}] and P[x_{1}ϵx_{2}],

• for every H such that P[H] holds P[¬H],

• for all H_{1}, H_{2} such that P[H_{1}] and P[H_{2}] holds P[H_{1} ∧ H_{2}],

• for all H, x such that P[H] holds P[∀_{x}H].

For simplicity we adopt the following rules: M, E will denote non-empty families of sets, e will denote an element of E, m, m′ will denote elements of M, f, g will denote functions from VAR into E, and v, v′ will denote functions from VAR into M. Let us consider E, f, x, e. The functor f(\frac{x}{e}) yields a function from VAR into E and is defined by: (Def.1) (f(\frac{x}{e}))(x) = e and for every y such that (f(\frac{x}{e}))(y) ≠ f(y) holds x = y.

The following proposition is true

(76) g = f(\frac{x}{e}) if and only if g(x) = e and for every y such that g(y) ≠ f(y)

holds x = y. Let D, D_{1}, D_{2} be non-empty sets, and let f be a function from D into D_{1}.

Let us assume that D_{1} ⊆ D_{2}. The functor D_{2}[f] yields a function from D into D_{2} and is defined as follows: (Def.2) D_{2}[f] = f.

Next we state several propositions:

(77) For all non-empty sets D, D_{1}, D_{2} and for every function f from D into D_{1} such that D_{1} ⊆ D_{2} holds D_{2}[f] = f.

(78) (v(\frac{x}{m′}))(\frac{x}{m}) = v(\frac{x}{m}) and v(\frac{x}{v(x)}) = v.

(79) If x ≠ y, then (v(\frac{x}{m}))(\frac{y}{m′}) = (v(\frac{y}{m′}))(\frac{x}{m}).

(80) M, v ⊧ ∀_{x}H if and only if for every m holds M, v(\frac{x}{m}) ⊧ H.

(81) M, v ⊧ ∀_{x}H if and only if M, v(\frac{x}{m}) ⊧ ∀_{x}H.

(82) M, v ⊧ ∃_{x}H if and only if there exists m such that M, v(\frac{x}{m}) ⊧ H.

(83) M, v ⊧ ∃_{x}H if and only if M, v(\frac{x}{m}) ⊧ ∃_{x}H.

(84) For all v, v′ such that for every x such that x ∈ Free H holds v′(x) = v(x) holds if M, v ⊧ H, then M, v′ ⊧ H.

(85) Free H is finite.

In the sequel i, j will denote natural numbers. The following propositions are true:

(86) If x_{i} = x_{j}, then i = j.

(87) There exists i such that x = x_{i}.

(88) x is a natural number and x ∈ .

(89) M, v ⊧ x=x.

(90) M ⊧ x=x.

(91) M, v |̸= xϵx.

(92) M |̸= xϵx and M ⊧ ¬xϵx.

(93) M ⊧ x=y if and only if x = y or there exists a such that {a} = M.

(94) M ⊧ ¬xϵy if and only if x = y or for every X such that X ∈ M holds X ∩ M = ∅.

(95) If H is an equality, then M, v ⊧ H if and only if v(Var_{1}(H)) = v(Var_{2}(H)).

(96) If H is a membership, then M, v ⊧ H if and only if v(Var_{1}(H)) ∈ v(Var_{2}(H)).

(97) If H is negative, then M, v ⊧ H if and only if M, v |̸= Arg(H).

(98) If H is conjunctive, then M, v ⊧ H if and only if M, v ⊧ LeftArg(H) and M, v ⊧ RightArg(H).

(99) If H is universal, then M, v ⊧ H if and only if for every m holds M, v(\frac{Bound(H)}{m}) ⊧ Scope(H).

(100) If H is disjunctive, then M, v ⊧ H if and only if M, v ⊧ LeftArg(H) or M, v ⊧ RightArg(H).

(101) If H is conditional, then M, v ⊧ H if and only if if M, v ⊧ Antecedent(H), then M, v ⊧ Consequent(H).

(102) If H is biconditional, then M, v ⊧ H if and only if M, v ⊧ LeftSide(H) if and only if M, v ⊧ RightSide(H).

(103) If H is existential, then M, v ⊧ H if and only if there exists m such that M, v(\frac{Bound(H)}{m}) ⊧ Scope(H).

(104) M ⊧ ∃_{x}H if and only if for every v there exists m such that M, v(\frac{x}{m}) ⊧ H.

(105) If M ⊧ H, then M ⊧ ∃_{x}H.

(106) M ⊧ H if and only if M ⊧ ∀_{x,y}H.

(107) If M ⊧ H, then M ⊧ ∃_{x,y}H.

(108) M ⊧ H if and only if M ⊧ ∀_{x,y,z}H.

(109) If M ⊧ H, then M ⊧ ∃_{x,y,z}H.

(110) M, v ⊧ (p ⇔ q) ⇒ (p ⇒ q) and M ⊧ (p ⇔ q) ⇒ (p ⇒ q).

(111) M, v ⊧ (p ⇔ q) ⇒ (q ⇒ p) and M ⊧ (p ⇔ q) ⇒ (q ⇒ p).

(112) M ⊧ (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒ r)).

(113) If M, v ⊧ p ⇒ q and M, v ⊧ q ⇒ r, then M, v ⊧ p ⇒ r.

(114) If M ⊧ p ⇒ q and M ⊧ q ⇒ r, then M ⊧ p ⇒ r.

(115) M, v ⊧ (p ⇒ q) ∧ (q ⇒ r) ⇒ (p ⇒ r) and M ⊧ (p ⇒ q) ∧ (q ⇒ r) ⇒ (p ⇒ r).

(116) M, v ⊧ p ⇒ (q ⇒ p) and M ⊧ p ⇒ (q ⇒ p).

(117) M, v ⊧ (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) and M ⊧ (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)).

(118) M, v ⊧ p ∧ q ⇒ p and M ⊧ p ∧ q ⇒ p.

(119) M, v ⊧ p ∧ q ⇒ q and M ⊧ p ∧ q ⇒ q.

(120) M, v ⊧ p ∧ q ⇒ q ∧ p and M ⊧ p ∧ q ⇒ q ∧ p.

(121) M, v ⊧ p ⇒ p ∧ p and M ⊧ p ⇒ p ∧ p.

(122) M, v ⊧ (p ⇒ q) ⇒ ((p ⇒ r) ⇒ (p ⇒ q ∧ r)) and M ⊧ (p ⇒ q) ⇒ ((p ⇒ r) ⇒ (p ⇒ q ∧ r)).

(123) M, v ⊧ p ⇒ p ∨ q and M ⊧ p ⇒ p ∨ q.

(124) M, v ⊧ q ⇒ p ∨ q and M ⊧ q ⇒ p ∨ q.

(125) M, v ⊧ p ∨ q ⇒ q ∨ p and M ⊧ p ∨ q ⇒ q ∨ p.

(126) M, v ⊧ p ⇒ p ∨ p and M ⊧ p ⇒ p ∨ p.

(127) M, v ⊧ (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) and M ⊧ (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)).

(128) M, v ⊧ (p ⇒ r) ∧ (q ⇒ r) ⇒ (p ∨ q ⇒ r) and M ⊧ (p ⇒ r) ∧ (q ⇒ r) ⇒ (p ∨ q ⇒ r).

(129) M, v ⊧ (p ⇒ ¬q) ⇒ (q ⇒ ¬p) and M ⊧ (p ⇒ ¬q) ⇒ (q ⇒ ¬p).

(130) M, v ⊧ ¬p ⇒ (p ⇒ q) and M ⊧ ¬p ⇒ (p ⇒ q).

(131) M, v ⊧ (p ⇒ q) ∧ (p ⇒ ¬q) ⇒ ¬p and M ⊧ (p ⇒ q) ∧ (p ⇒ ¬q) ⇒ ¬p.

(132) If M, v ⊧ p ⇒ q and M, v ⊧ p, then M, v ⊧ q.

(133) If M ⊧ p ⇒ q and M ⊧ p, then M ⊧ q.

(134) M, v ⊧ ¬(p ∧ q) ⇒ ¬p ∨ ¬q and M ⊧ ¬(p ∧ q) ⇒ ¬p ∨ ¬q.

(135) M, v ⊧ ¬p ∨ ¬q ⇒ ¬(p ∧ q) and M ⊧ ¬p ∨ ¬q ⇒ ¬(p ∧ q).

(136) M, v ⊧ ¬(p ∨ q) ⇒ ¬p ∧ ¬q and M ⊧ ¬(p ∨ q) ⇒ ¬p ∧ ¬q.

(137) M, v ⊧ ¬p ∧ ¬q ⇒ ¬(p ∨ q) and M ⊧ ¬p ∧ ¬q ⇒ ¬(p ∨ q).

(138) M ⊧ (∀_{x}H) ⇒ H.

(139) M ⊧ H ⇒ (∃_{x}H).

(140) If x ∉ Free H_{1}, then M ⊧ (∀_{x}H_{1} ⇒ H_{2}) ⇒ (H_{1} ⇒ (∀_{x}H_{2})).

(141) If x ∉ Free H_{1} and M ⊧ H_{1} ⇒ H_{2}, then M ⊧ H_{1} ⇒ (∀_{x}H_{2}).

(142) If x ∉ Free H_{2}, then M ⊧ (∀_{x}H_{1} ⇒ H_{2}) ⇒ ((∃_{x}H_{1}) ⇒ H_{2}).

(143) If x ∉ Free H_{2} and M ⊧ H_{1} ⇒ H_{2}, then M ⊧ (∃_{x}H_{1}) ⇒ H_{2}.

(144) If M ⊧ H_{1} ⇒ (∀_{x}H_{2}), then M ⊧ H_{1} ⇒ H_{2}.

(145) If M ⊧ (∃_{x}H_{1}) ⇒ H_{2}, then M ⊧ H_{1} ⇒ H_{2}.

(146) WFF ⊆ 2^{[: , :]} .

Let us consider H. The functor Var_{H} yields a set and is defined by: (Def.3) Var_{H} = rng H \ {0, 1, 2, 3, 4}.

We now state a number of propositions:

(147) Var_{H} = rng H \ {0, 1, 2, 3, 4}.

(148) x ≠ 0 and x ≠ 1 and x ≠ 2 and x ≠ 3 and x ≠ 4.

(149) x ∉ {0, 1, 2, 3, 4}.

(150) If a ∈ Var_{H}, then a ≠ 0 and a ≠ 1 and a ≠ 2 and a ≠ 3 and a ≠ 4.

(151) Var_{x=y} = {x, y}.

(152) Var_{xϵy} = {x, y}.

(153) Var_{¬H} = Var_{H}.

(154) Var_{H_{1}∧H_{2}} = Var_{H_{1}} ∪ Var_{H_{2}}.

(155) Var_{∀_{x}H} = Var_{H} ∪ {x}.

(156) Var_{H_{1}∨H_{2}} = Var_{H_{1}} ∪ Var_{H_{2}}.

(157) Var_{H_{1}⇒H_{2}} = Var_{H_{1}} ∪ Var_{H_{2}}.

(158) Var_{H_{1}⇔H_{2}} = Var_{H_{1}} ∪ Var_{H_{2}}.

(159) Var_{∃_{x}H} = Var_{H} ∪ {x}.

(160) Var_{∀_{x,y}H} = Var_{H} ∪ {x, y}.

(161) Var_{∃_{x,y}H} = Var_{H} ∪ {x, y}.

(162) Var_{∀_{x,y,z}H} = Var_{H} ∪ {x, y, z}.

(163) Var_{∃_{x,y,z}H} = Var_{H} ∪ {x, y, z}.

(164) Free H ⊆ Var_{H}.

Let us consider H. Then Var_{H} is a non-empty subset of VAR.

Let us consider H, x, y. The functor H(\frac{x}{y}) yields a function and is defined by: (Def.4) dom(H(\frac{x}{y})) = dom H and for every a such that a ∈ dom H holds if

H(a) = x, then (H(\frac{x}{y}))(a) = y but if H(a) ≠ x, then (H(\frac{x}{y}))(a) = H(a). One can prove the following propositions:

(165) For every function f holds f = H(\frac{x}{y}) if and only if dom f = dom H and for every a such that a ∈ dom H holds if H(a) = x, then f(a) = y but if H(a) ≠ x, then f(a) = H(a).

(166) x_{1}=x_{2}(\frac{y_{1}}{y_{2}} ) = z_{1}=z_{2} if and only if x_{1} ≠ y_{1} and x_{2} ≠ y_{1} and z_{1} = x_{1} and z_{2} = x_{2} or x_{1} = y_{1} and x_{2} ≠ y_{1} and z_{1} = y_{2} and z_{2} = x_{2} or x_{1} ≠ y_{1} and x_{2} = y_{1} and z_{1} = x_{1} and z_{2} = y_{2} or x_{1} = y_{1} and x_{2} = y_{1} and z_{1} = y_{2} and z_{2} = y_{2}.

(167) There exist z_{1}, z_{2} such that x_{1}=x_{2}(\frac{y_{1}}{y_{2}} ) = z_{1}=z_{2}.

(168) x_{1}ϵx_{2}(\frac{y_{1}}{y_{2}} ) = z_{1}ϵz_{2} if and only if x_{1} ≠ y_{1} and x_{2} ≠ y_{1} and z_{1} = x_{1} and z_{2} = x_{2} or x_{1} = y_{1} and x_{2} ≠ y_{1} and z_{1} = y_{2} and z_{2} = x_{2} or x_{1} ≠ y_{1} and x_{2} = y_{1} and z_{1} = x_{1} and z_{2} = y_{2} or x_{1} = y_{1} and x_{2} = y_{1} and z_{1} = y_{2} and z_{2} = y_{2}.

(169) There exist z_{1}, z_{2} such that x_{1}ϵx_{2}(\frac{y_{1}}{y_{2}} ) = z_{1}ϵz_{2}.

(170) ¬F = (¬H)(\frac{x}{y}) if and only if F = H(\frac{x}{y}).

(171) H(\frac{x}{y}) ∈ WFF.

Let us consider H, x, y. Then H(\frac{x}{y}) is a ZF-formula.

The following propositions are true:

(172) G_{1} ∧ G_{2} = (H_{1} ∧ H_{2})(\frac{x}{y}) if and only if G_{1} = H_{1}(\frac{x}{y}) and G_{2} = H_{2}(\frac{x}{y}).

(173) If z ≠ x, then ∀_{z}G = (∀_{z}H)(\frac{x}{y}) if and only if G = H(\frac{x}{y}).

(174) ∀_{y}G = (∀_{x}H)(\frac{x}{y}) if and only if G = H(\frac{x}{y}).

(175) G_{1} ∨ G_{2} = (H_{1} ∨ H_{2})(\frac{x}{y}) if and only if G_{1} = H_{1}(\frac{x}{y}) and G_{2} = H_{2}(\frac{x}{y}).

(176) G_{1} ⇒ G_{2} = (H_{1} ⇒ H_{2})(\frac{x}{y}) if and only if G_{1} = H_{1}(\frac{x}{y}) and G_{2} = H_{2}(\frac{x}{y}).

(177) G_{1} ⇔ G_{2} = (H_{1} ⇔ H_{2})(\frac{x}{y}) if and only if G_{1} = H_{1}(\frac{x}{y}) and G_{2} = H_{2}(\frac{x}{y}).

(178) If z ≠ x, then ∃_{z}G = (∃_{z}H)(\frac{x}{y}) if and only if G = H(\frac{x}{y}).

(179) ∃_{y}G = (∃_{x}H)(\frac{x}{y}) if and only if G = H(\frac{x}{y}).

(180) H is an equality if and only if H(\frac{x}{y}) is an equality.

(181) H is a membership if and only if H(\frac{x}{y}) is a membership.

(182) H is negative if and only if H(\frac{x}{y}) is negative.

(183) H is conjunctive if and only if H(\frac{x}{y}) is conjunctive.

(184) H is universal if and only if H(\frac{x}{y}) is universal.

(185) If H is negative, then Arg(H(\frac{x}{y})) = Arg(H)(\frac{x}{y}).

(186) If H is conjunctive, then LeftArg(H(\frac{x}{y})) = LeftArg(H)(\frac{x}{y}) and RightArg(H(\frac{x}{y})) = RightArg(H)(\frac{x}{y}).

(187) If H is universal, then Scope(H(\frac{x}{y})) = Scope(H)(\frac{x}{y}) but if Bound(H) = x, then Bound(H(\frac{x}{y})) = y but if Bound(H) ≠ x, then Bound(H(\frac{x}{y})) = Bound(H).

(188) H is disjunctive if and only if H(\frac{x}{y}) is disjunctive.

(189) H is conditional if and only if H(\frac{x}{y}) is conditional.

(190) If H is biconditional, then H(\frac{x}{y}) is biconditional.

(191) H is existential if and only if H(\frac{x}{y}) is existential.

(192) If H is disjunctive, then LeftArg(H(\frac{x}{y})) = LeftArg(H)(\frac{x}{y}) and RightArg(H(\frac{x}{y})) = RightArg(H)(\frac{x}{y}).

(193) If H is conditional, then Antecedent(H(\frac{x}{y})) = Antecedent(H)(\frac{x}{y}) and Consequent(H(\frac{x}{y})) = Consequent(H)(\frac{x}{y}).

(194) If H is biconditional, then LeftSide(H(\frac{x}{y})) = LeftSide(H)(\frac{x}{y}) and RightSide(H(\frac{x}{y})) = RightSide(H)(\frac{x}{y}).

(195) If H is existential, then Scope(H(\frac{x}{y})) = Scope(H)(\frac{x}{y}) but if Bound(H) = x, then Bound(H(\frac{x}{y})) = y but if Bound(H) ≠ x, then Bound(H(\frac{x}{y})) = Bound(H).

(196) If x ∉ Var_{H}, then H(\frac{x}{y}) = H.

(197) H(\frac{x}{x}) = H.

(198) If x ≠ y, then x ∉ Var_{H( \frac{x}{y})}.

(199) If x ∈ Var_{H}, then y ∈ Var_{H( \frac{x}{y})}.

(200) If x ≠ y, then (H(\frac{x}{y}))(\frac{x}{z}) = H(\frac{x}{y}).

(201) Var_{H( \frac{x}{y})} ⊆ (Var_{H} \ \{x\}) ∪ \{y\}.

References

[1] Grzegorz Bancerek. A model of ZF set theory language. Formalized Mathematics, 1(1):131–145, 1990.

[2] Grzegorz Bancerek. Models and satisfiability. Formalized Mathematics, 1(1):191–199, 1990.

[3] Grzegorz Bancerek. Properties of zf models. Formalized Mathematics, 1(2):277–280, 1990.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[8] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 972 Grzegorz Bancerek

[9] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathemat-

ics, 1(1):9–11, 1990. Received August 10, 1990

The Reflection Theorem

Grzegorz Bancerek^{1} Warsaw University Białystok

Summary. The goal is show that the reflection theorem holds. The theorem is as usual in the Morse-Kelley theory of classes (MK). That theory works with universal class which consists of all sets and every class is a subclass of it. In this paper (and in another Mizar articles) we work in Tarski-Grothendieck (TG) theory (see [16]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [14] and some concepts of MK are modeled. The concepts are: the class On of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of non-empty elements of universe, etc. The reflection theorem states that if A_{ξ} is an increasing and continuous transfinite sequence of non-empty sets and class A = ⋃_{ξ∈On} A_{ξ}, then for every formula H there is a strictly increasing continuous mapping F : On → On such that if is a critical number of F (i.e. F ( ) = > 0) and f ∈ AVAR_{}, then A, f ⊧ H ≡ A_{}, f ⊧ H. The proof is based on [13]. Besides, in the article it is shown that every universal class is a model of ZF set theory if ω (the first infinite ordinal number) belongs to it. Some propositions concerning ordinal numbers and sequences of them are also present.

MML Identifier: ZF REFLE.

The notation and terminology used in this paper have been introduced in the following articles: [16], [15], [11], [12], [4], [5], [6], [10], [8], [1], [3], [9], [14], [2], and [7]. In the sequel W is a universal class, H is a ZF-formula, x is arbitrary, and X is a set. We now state several propositions:

(1) W ⊧ the axiom of extensionality.

(2) W ⊧ the axiom of pairs.

(3) W ⊧ the axiom of unions.

(4) If ω ∈ W, then W ⊧ the axiom of infinity.

(5) W ⊧ the axiom of power sets.

1Supported by RPBP III-24.C1.

(6) For every H such that {x_{0}, x_{1}, x_{2}} misses Free H holds W ⊧ the axiom of substitution for H.

(7) If ω ∈ W, then W is a model of ZF.

For simplicity we follow the rules: E denotes a non-empty family of sets, F denotes a function, f denotes a function from VAR into E, A, B, C denote ordinal numbers, a, b denote ordinals of W, p_{1} denotes a transfinite sequence of ordinals of W, and H denotes a ZF-formula. Let us consider A, B. Let us note that one can characterize the predicate A ⊆ B by the following (equivalent) condition: (Def.1) for every C such that C ∈ A holds C ∈ B.

In this article we present several logical schemes. The scheme ALFA deals with a non-empty set A, and a binary predicate P, and states that:

there exists F such that dom F = A and for every element d of A there exists A such that A = F (d) and P[d, A] and for every B such that P[d, B] holds A ⊆ B provided the parameters meet the following condition:

• for every element d of A there exists A such that P[d, A].

The scheme ALFA’Universe deals with a universal class A, a non-empty set B, and a binary predicate P, and states that:

there exists F such that dom F = B and for every element d of B there exists an ordinal a of A such that a = F (d) and P[d, a] and for every ordinal b of A such that P[d, b] holds a ⊆ b provided the following condition is met:

• for every element d of B there exists an ordinal a of A such that P[d, a].

One can prove the following proposition

(8) x is an ordinal of W if and only if x ∈ On W.

In the sequel p_{2} is a sequence of ordinal numbers. Now we present three schemes. The scheme OrdSeqOfUnivEx deals with a universal class A, and a binary predicate P, and states that:

there exists a transfinite sequence p_{1} of ordinals of A such that for every ordinal a of A holds P[a, p_{1}(a)] provided the following conditions are satisfied:

• for all ordinals a, b_{1}, b_{2} of A such that P[a, b_{1}] and P[a, b_{2}] holds b_{1} = b_{2},

• for every ordinal a of A there exists an ordinal b of A such that P[a, b].

The scheme UOS Exist concerns a universal class A, an ordinal B of A, a binary functor F yielding an ordinal of A, and a binary functor G yielding an ordinal of A and states that:

there exists a transfinite sequence p_{1} of ordinals of A such that p_{1}(0_{A}) = B and for all ordinals a, b of A such that b = p_{1}(a) holds p_{1}(succ a) = F(a, b) and for every ordinal a of A and for every sequence p_{2} of ordinal numbers such that a ≠ 0_{A} and a is a limit ordinal number and p_{2} = p_{1} a holds p_{1}(a) = G(a, p_{2}) for all values of the parameters.

The scheme Universe Ind concerns a universal class A, and a unary predicate P, and states that:

for every ordinal a of A holds P[a] provided the parameters have the following properties:

• P[0_{A}],

• for every ordinal a of A such that P[a] holds P[succ a],

• for every ordinal a of A such that a ≠ 0_{A} and a is a limit ordinal number and for every ordinal b of A such that b ∈ a holds P[b] holds P[a].

Let f be a function, and let W be a universal class, and let a be an ordinal of W. The functor ⋃_{a} f yields a set and is defined as follows: (Def.2) ⋃_{a} f = ⋃(W (f R_{a})).

We now state several propositions:

(9) ⋃_{a} f = ⋃(W (f R_{a})).

(10) For every transfinite sequence L and for every A holds L R_{A} is a transfinite sequence.

(11) For every sequence L of ordinal numbers and for every A holds L R_{A} is a sequence of ordinal numbers.

(12) ⋃ p_{2} is an ordinal number.

(13) ⋃(X p_{2}) is an ordinal number.

(14) On R_{A} = A.

(15) p_{2} R_{A} = p_{2} A.

Let p_{1} be a sequence of ordinal numbers, and let W be a universal class, and let a be an ordinal of W. Then ⋃_{a} p_{1} is an ordinal of W.

Next we state the proposition (17)^{2} For every transfinite sequence p_{1} of ordinals of W holds ⋃_{a} p_{1} = ⋃(p_{1}

a) and ⋃_{a}(p_{1} a) = ⋃(p_{1} a). Let W be a universal class, and let a, b be ordinals of W. Then a ∪ b is an

ordinal of W.

Let us consider W. A non-empty family of sets is said to be a non-empty set from W if: (Def.3) it ∈ W.

Let us consider W. A non-empty family of sets is said to be a subclass of W if: (Def.4) it ⊆ W.

Let us consider W. A transfinite sequence of elements of W is called a transfinite sequence of non-empty sets from W if: (Def.5) dom it = On W and ∅ ∉ rng it.

2The proposition (16) became obvious.

We now state four propositions:

(18) E is a non-empty set from W if and only if E ∈ W.

(19) E is a subclass of W if and only if E ⊆ W.

(20) For every transfinite sequence T of elements of W holds T is a transfinite sequence of non-empty sets from W if and only if dom T = On W and ∅ ∉ rng T.

(21) For every non-empty set D from W holds D is a subclass of W. Let us consider W, and let L be a transfinite sequence of non-empty sets

from W. Then ⋃ L is a subclass of W. Let us consider a. Then L(a) is a non-empty set from W.

In the sequel L is a transfinite sequence of non-empty sets from W and f is a function from VAR into L(a). Next we state several propositions:

(22) If X ∈ W, then \overline{\overline{X}} < \overline{\overline{W}}.

(23) a ∈ dom L.

(24) L(a) ⊆ ⋃ L.

(25) ≈ VAR and \overline{\overline{VAR}} = \overline{\overline{}}.

(26) ⋃(On X) is an ordinal number.

(27) sup X ⊆ succ(⋃(On X)).

(28) If X ∈ W, then sup X ∈ W.

The following proposition is true

(29) Suppose ω ∈ W and for all a, b such that a ∈ b holds L(a) ⊆ L(b) and for every a such that a ≠ 0 and a is a limit ordinal number holds L(a) = ⋃(L a). Then for every H there exists p_{1} such that p_{1} is increasing and p_{1} is continuous and for every a such that p_{1}(a) = a and 0 ≠ a for every f holds ⋃ L, ⋃ L[f] ⊧ H if and only if L(a), f ⊧ H.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711–714, 1990.

[3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589– 593, 1990.

[4] Grzegorz Bancerek. A model of ZF set theory language. Formalized Mathematics, 1(1):131–145, 1990.

[5] Grzegorz Bancerek. Models and satisfiability. Formalized Mathematics, 1(1):191–199, 1990.

[6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[7] Grzegorz Bancerek. Replacing of variables in formulas of ZF theory. Formalized Mathematics, 1(5):963–972, 1990.

[8] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281–290, 1990.

[9] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.

[10] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

[11] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[12] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[13] Andrzej Mostowski. Constructible Sets with Applications. North Holland, 1969.

[14] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

[15] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

978

Binary Operations on Finite Sequences

Wojciech A. Trybulec^{1} Warsaw University

Summary. We generalize the semigroup operation on finite sequences introduced in [6] for binary operations that have a unity or for non-empty finite sequences.

MML Identifier: FINSOP 1.

The papers [9], [4], [5], [2], [3], [8], [6], [7], and [1] provide the notation and terminology for this paper. For simplicity we adopt the following convention: D denotes a non-empty set, d, d_{1}, d_{2}, d_{3} denote elements of D, F, G, H denote finite sequences of elements of D, f denotes a function from into D, g denotes a binary operation on D, k, n, l denote natural numbers, and P denotes a permutation of Seg(len F). Let us consider D, n, d. Then n ⟼ d is a finite sequence of elements of D.

Let us consider D, F, g. Let us assume that g has a unity or len F ≥ 1. The functor g ⊙ F yields an element of D and is defined by: (Def.1) g ⊙ F = 1_{g} if g has a unity and len F = 0, there exists f such that

f(1) = F (1) and for every n such that 0 ≠ n and n < len F holds f(n + 1) = g(f(n), F (n + 1)) and g ⊙ F = f(len F), otherwise.

One can prove the following propositions:

(1) If g has a unity and len F = 0, then g ⊙ F = 1_{g}.

(2) Suppose len F ≥ 1. Then there exists f such that f(1) = F (1) and for every n such that 0 ≠ n and n < len F holds f(n+1) = g(f(n), F (n+1)) and g ⊙ F = f(len F).

(3) Suppose len F ≥ 1 and there exists f such that f(1) = F (1) and for every n such that 0 ≠ n and n < len F holds f(n+1) = g(f(n), F (n+1)) and d = f(len F). Then d = g ⊙ F.

(4) If g has a unity or len F ≥ 1 but g is associative and g is commutative, then g ⊙ F = g F.

1Supported by RPBP.III-24.C1

(5) If g has a unity or len F ≥ 1, then g ⊙ F ^{}⟨d⟩ = g(g ⊙ F, d).

(6) If g is associative but g has a unity or len F ≥ 1 and len G ≥ 1, then g ⊙ F ^{}G = g(g ⊙ F, g ⊙ G).

(7) If g is associative but g has a unity or len F ≥ 1, then g ⊙ ⟨d⟩^{}F = g(d, g ⊙ F).

(8) If g is commutative and g is associative but g has a unity or len F ≥ 1 and G = F · P, then g ⊙ F = g ⊙ G.

(9) If g has a unity or len F ≥ 1 but g is associative and g is commutative and F is one-to-one and G is one-to-one and rng F = rng G, then g ⊙ F = g ⊙ G.

(10) Suppose g is associative and g is commutative but g has a unity or len F ≥ 1 and len F = len G and len F = len H and for every k such that k ∈ Seg(len F) holds F (k) = g(G(k), H(k)). Then g ⊙ F = g(g ⊙ G, g ⊙ H).

(11) If g has a unity, then g ⊙ ε_{D} = 1_{g}.

(12) g ⊙ ⟨d⟩ = d.

(13) g ⊙ ⟨d_{1}, d_{2}⟩ = g(d_{1}, d_{2}).

(14) If g is commutative, then g ⊙ ⟨d_{1}, d_{2}⟩ = g ⊙ ⟨d_{2}, d_{1}⟩.

(15) g ⊙ ⟨d_{1}, d_{2}, d_{3}⟩ = g(g(d_{1}, d_{2}), d_{3}).

(16) If g is commutative, then g ⊙ ⟨d_{1}, d_{2}, d_{3}⟩ = g ⊙ ⟨d_{2}, d_{1}, d_{3}⟩.

(17) g ⊙ (1 ⟼ d) = d.

(18) g ⊙ (2 ⟼ d) = g(d, d).

(19) If g is associative but g has a unity or k ≠ 0 and l ≠ 0, then g⊙(k+l ⟼ d) = g(g ⊙ (k ⟼ d), g ⊙ (l ⟼ d)).

(20) If g is associative but g has a unity or k ≠ 0 and l ≠ 0, then g ⊙(k·l ⟼ d) = g ⊙ (l ⟼ g ⊙ (k ⟼ d)).

(21) If len F = 1, then g ⊙ F = F (1).

(22) If len F = 2, then g ⊙ F = g(F (1), F (2)).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Semigroup operations on finite subsets. Formalized Mathematics, 1(4):651–656, 1990.

[7] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[8] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

[9] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

982

Finite Join and Finite Meet, and Dual Lattices

Andrzej Trybulec^{1} Warsaw University Białystok

Summary. The concepts of finite join and finite meet in a lattice are introduced. Some properties of the finite join are proved. After introducing the concept of dual lattice in view of dualism we obtain analogous properties of the meet. We prove these properties of binary operations in a lattice, which are usually included in axioms of the lattice theory. We also introduce the concept of Heyting lattice (a bounded lattice with relative pseudo-complements).

MML Identifier: LATTICE2.

The papers [10], [3], [4], [5], [8], [2], [11], [6], [9], [7], and [1] provide the notation and terminology for this paper. For simplicity we adopt the following convention: A denotes a set, C denotes a non-empty set, B denotes a subset of A, x denotes an element of A, and f, g denote functions from A into C. The following propositions are true:

(1) f B is a function from B into C.

(2) dom(g B) = B.

(3) f ^{∘} B = (f B)^{∘} B.

(4) If x ∈ B, then (f B)(x) = f(x).

(5) f B = g B if and only if for every x such that x ∈ B holds g(x) = f(x).

(6) For every set B holds f +· g B is a function from A into C.

(7) g B +· f = f.

(8) For all functions f, g such that g ≤ f holds f +· g = f.

(9) f +· f B = f.

(10) If for every x such that x ∈ B holds g(x) = f(x), then f +· g B = f.

1Supported by RPBP.III-24.C1.

In the sequel B will denote a finite subset of A. We now state several propositions:

(11) For every set X holds X is a finite subset of A if and only if X ⊆ A and X is finite.

(12) g B +· f = f.

(13) dom(g B) = B.

(14) If for every x such that x ∈ B holds g(x) = f(x), then f +· g B = f.

(15) f ^{∘} B = (f B)^{∘} B.

(16) If f B = g B, then f ^{∘} B = g^{∘} B.

Let D be a non-empty set, and let o, o′ be binary operations on D. We say that o absorbs o′ if and only if: (Def.1) for all elements x, y of D holds o(x, o′(x, y)) = x.

In the sequel L will be a lattice structure. The following proposition is true (17) If the join operation of L is commutative and the join operation of L

is associative and the meet operation of L is commutative and the meet operation of L is associative and the join operation of L absorbs the meet operation of L and the meet operation of L absorbs the join operation of L, then L is a lattice.

Let L be a lattice structure. The functor L^{∘} yields a lattice structure and is defined by: (Def.2) L^{∘} = ⟨ the carrier of L, the meet operation of L, the join operation of

L⟩. One can prove the following propositions:

(18) The carrier of L = the carrier of L^{∘} and the join operation of L = the meet operation of L^{∘} and the meet operation of L = the join operation of L^{∘}.

(19) (L^{∘})^{∘} = L.

We follow the rules: L will be a lattice and a, b, u, v will be elements of the carrier of L. We now state a number of propositions:

(20) If for every v holds u ⊓ v = u, then u = ⊥_{L}.

(21) If for every v holds u ⊔ v = v, then u = ⊥_{L}.

(22) If for every v holds (the join operation of L)(u, v) = v, then u = ⊥_{L}.

(23) If for every v holds u ⊔ v = u, then u = ⊤_{L}.

(24) If for every v holds u ⊓ v = v, then u = ⊤_{L}.

(25) If for every v holds (the meet operation of L)(u, v) = v, then u = ⊤_{L}.

(26) The join operation of L is idempotent.

(27) The join operation of L is commutative.

(28) If the join operation of L has a unity, then ⊥_{L} = 1_{the join operation of L}.

(29) The join operation of L is associative.

(30) The meet operation of L is idempotent.

(31) The meet operation of L is commutative.

(32) The meet operation of L is associative.

(33) If the meet operation of L has a unity, then ⊤_{L} = 1_{the meet operation of L}.

(34) The join operation of L is distributive w.r.t. the join operation of L.

(35) If L is a distributive lattice, then the join operation of L is distributive w.r.t. the meet operation of L.

(36) If the join operation of L is distributive w.r.t. the meet operation of L, then L is a distributive lattice.

(37) If L is a distributive lattice, then the meet operation of L is distributive w.r.t. the join operation of L.

(38) If the meet operation of L is distributive w.r.t. the join operation of L, then L is a distributive lattice.

(39) The meet operation of L is distributive w.r.t. the meet operation of L.

(40) The join operation of L absorbs the meet operation of L.

(41) The meet operation of L absorbs the join operation of L.

We now define two new functors. Let A be a non-empty set, and let L be a lattice, and let B be a finite subset of A, and let f be a function from A into the carrier of L. The functor ⨆_{B}^{f} f yields an element of the carrier of L and is defined as follows: (Def.3) ⨆_{B}^{f} f = (the join operation of L)- ∑_{B} f. The functor ⌈−⌉_{B}^{f}f yields an element of the carrier of L and is defined by: (Def.4) ⌈−⌉_{B}^{f}f = (the meet operation of L)- ∑_{B} f.

We now state the proposition

(42) For every non-empty set A and for every lattice L and for every finite subset B of A and for every function f from A into the carrier of L holds ⨆_{B}^{f} f = (the join operation of L)- ∑_{B} f.

For simplicity we adopt the following convention: A will be a non-empty set, x will be an element of A, B will be a finite subset of A, and f, g will be functions from A into the carrier of L. Next we state several propositions:

(43) If x ∈ B, then f(x) ⊑ ⨆_{B}^{f} f.

(44) If there exists x such that x ∈ B and u ⊑ f(x), then u ⊑ ⨆_{B}^{f} f.

(45) If for every x such that x ∈ B holds f(x) = u and B ≠ ∅, then ⨆_{B}^{f} f = u.

(46) If ⨆_{B}^{f} f ⊑ u, then for every x such that x ∈ B holds f(x) ⊑ u.

(47) If B ≠ ∅ and for every x such that x ∈ B holds f(x) ⊑ u, then ⨆_{B}^{f} f ⊑ u.

(48) If B ≠ ∅ and for every x such that x ∈ B holds f(x) ⊑ g(x), then ⨆_{B}^{f} f ⊑ ⨆_{B}^{f} g.

(49) If B ≠ ∅ and f B = g B, then ⨆_{B}^{f} f = ⨆_{B}^{f} g.

(50) If B ≠ ∅, then v ⊔ ⨆_{B}^{f} f = ⨆_{B}^{f}( (the join operation of L)^{∘}(v, f)). Let L be a lattice. Then L^{∘} is a lattice. We now state a number of propositions:

(51) For every lattice L and for every finite subset B of A and for every function f from A into the carrier of L and for every function f′ from A into the carrier of L^{∘} such that f = f′ holds ⨆_{B}^{f} f = ⌈−⌉_{B}^{f}f′ and ⌈−⌉_{B}^{f}f = ⨆_{B}^{f} f′.

(52) For all elements a′, b′ of the carrier of L^{∘} such that a = a′ and b = b′ holds a ⊓ b = a′ ⊔ b′ and a ⊔ b = a′ ⊓ b′.

(53) If a ⊑ b, then for all elements a′, b′ of the carrier of L^{∘} such that a = a′ and b = b′ holds b′ ⊑ a′.

(54) For all elements a′, b′ of the carrier of L^{∘} such that a′ ⊑ b′ and a = a′ and b = b′ holds b ⊑ a.

(55) If x ∈ B, then ⌈−⌉_{B}^{f}f ⊑ f(x).

(56) If there exists x such that x ∈ B and f(x) ⊑ u, then ⌈−⌉_{B}^{f}f ⊑ u.

(57) If for every x such that x ∈ B holds f(x) = u and B ≠ ∅, then ⌈−⌉_{B}^{f}f = u.

(58) If B ≠ ∅, then v ⊓ ⌈−⌉_{B}^{f}f = ⌈−⌉_{B}^{f}( (the meet operation of L)^{∘}(v, f)).

(59) If u ⊑ ⌈−⌉_{B}^{f}f, then for every x such that x ∈ B holds u ⊑ f(x).

(60) If B ≠ ∅ and f B = g B, then ⌈−⌉_{B}^{f}f = ⌈−⌉_{B}^{f}g.

(61) If B ≠ ∅ and for every x such that x ∈ B holds u ⊑ f(x), then u ⊑ ⌈−⌉_{B}^{f}f.

(62) If B ≠ ∅ and for every x such that x ∈ B holds f(x) ⊑ g(x), then ⌈−⌉_{B}^{f}f ⊑ ⌈−⌉_{B}^{f}g.

(63) For every lattice L holds L is a lower bound lattice if and only if L^{∘} is an upper bound lattice.

(64) For every lattice L holds L is an upper bound lattice if and only if L^{∘} is a lower bound lattice.

(65) L is a distributive lattice if and only if L^{∘} is a distributive lattice. In the sequel L denotes a lower bound lattice, f, g denote functions from A

into the carrier of L, and u denotes an element of the carrier of L. The following propositions are true:

(66) ⊥_{L} is a unity w.r.t. the join operation of L.

(67) The join operation of L has a unity.

(68) ⊥_{L} = 1_{the join operation of L}.

(69) If f B = g B, then ⨆_{B}^{f} f = ⨆_{B}^{f} g.

(70) If for every x such that x ∈ B holds f(x) ⊑ u, then ⨆_{B}^{f} f ⊑ u.

(71) If for every x such that x ∈ B holds f(x) ⊑ g(x), then ⨆_{B}^{f} f ⊑ ⨆_{B}^{f} g. In the sequel L will denote an upper bound lattice, f, g will denote functions

from A into the carrier of L, and u will denote an element of the carrier of L. The following propositions are true:

(72) ⊤_{L} is a unity w.r.t. the meet operation of L.

(73) The meet operation of L has a unity.

(74) ⊤_{L} = 1_{the meet operation of L}.

(75) If f B = g B, then ⌈−⌉_{B}^{f}f = ⌈−⌉_{B}^{f}g.

(76) If for every x such that x ∈ B holds u ⊑ f(x), then u ⊑ ⌈−⌉_{B}^{f}f.

(77) If for every x such that x ∈ B holds f(x) ⊑ g(x), then ⌈−⌉_{B}^{f}f ⊑ ⌈−⌉_{B}^{f}g.

(78) For every lower bound lattice L holds ⊥_{L} = ⊤_{L^{∘}}.

(79) For every upper bound lattice L holds ⊤_{L} = ⊥_{L^{∘}}.

A lower bound lattice is called a distributive lower bounded lattice if: (Def.5) it is a distributive lattice.

In the sequel L will denote a distributive lower bounded lattice, f, g will denote functions from A into the carrier of L, and u will denote an element of the carrier of L. We now state four propositions:

(80) The meet operation of L is distributive w.r.t. the join operation of L.

(81) (the meet operation of L)(u, ⨆_{B}^{f} f) = ⨆_{B}^{f}( (the meet operation of L)^{∘}(u, f)).

(82) If for every x such that x ∈ B holds g(x) = u ⊓ f(x), then u ⊓ ⨆_{B}^{f} f = ⨆_{B}^{f} g.

(83) u ⊓ ⨆_{B}^{f} f = ⨆_{B}^{f}( (the meet operation of L)^{∘}(u, f)).

A lower bound lattice is said to be a Heyting lattice if: (Def.6) it is a implicative lattice.

Next we state the proposition

(84) For every lower bound lattice L holds L is a Heyting lattice if and only if for every elements x, z of the carrier of L there exists an element y of the carrier of L such that x ⊓ y ⊑ z and for every element v of the carrier of L such that x ⊓ v ⊑ z holds v ⊑ y.

Let L be a lattice. We say that L is finite if and only if: (Def.7) the carrier of L is finite.

We now state several propositions:

(85) For every lattice L holds L is finite if and only if L^{∘} is finite.

(86) For every lattice L such that L is finite holds L is a lower bound lattice.

(87) For every lattice L such that L is finite holds L is an upper bound lattice.

(88) For every lattice L such that L is finite holds L is a bound lattice.

(89) For every distributive lattice L such that L is finite holds L is a Heyting lattice.

References

[1] Grzegorz Bancerek. Filters - part I. Formalized Mathematics, 1(5):813– 819, 1990.

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175– 180, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[7] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[8] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[9] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

[10] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[11] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990.

Consequences of the Reflection Theorem

Grzegorz Bancerek^{1} Warsaw University Białystok

Summary. Some consequences of the reflection theorem are discussed. To formulate them the notions of elementary equivalence and subsystems, and of models for a set of formulae are introduced. Besides, the concept of cofinality of a ordinal number with second one is used. The consequences of the reflection theorem (it is sometimes called the Scott-Scarpellini lemma) are: (i) If A_{ξ} is a transfinite sequence as in the reflection theorem (see [9]) and A = ⋃_{ξ∈On} A_{ξ}, then there is an increasing and continuous mapping ϕ from On into On such that for every critical number κ the set A_{κ} is an elementary subsystem of A (A_{κ} ≺ A). (ii) There is an increasing continuous mapping ϕ : On → On such that R_{κ} ≺ V for each of its critical numbers κ (V is the universal class and On is the class of all ordinals belonging to V). (iii) There are ordinal numbers α cofinal with ω for which R_{α} are models of ZF set theory. (iv) For each set X from universe V there is a model of ZF M which belongs to V and has X as an element.

MML Identifier: ZFREFLE1.

The articles [18], [14], [15], [19], [17], [8], [13], [5], [6], [1], [11], [4], [2], [7], [12], [16], [3], [10], and [9] provide the terminology and notation for this paper. We follow a convention: H, S will be ZF-formulae, X, Y will be sets, and e, u will be arbitrary. Let M be a non-empty family of sets, and let F be a subset of WFF. The predicate M ⊧ F is defined by: (Def.1) for every H such that H ∈ F holds M ⊧ H.

We now define two new predicates. Let M_{1}, M_{2} be non-empty families of sets. The predicate M_{1} ≡ M_{2} is defined as follows: (Def.2) for every H such that Free H = ∅ holds M_{1} ⊧ H if and only if M_{2} ⊧ H. Let us notice that this predicate is reflexive and symmetric. The predicate M_{1} ≺ M_{2} is defined as follows:

1Supported by RPBP III-24.C1.

(Def.3) M_{1} ⊆ M_{2} and for every H and for every function v from VAR into M_{1}

holds M_{1}, v ⊧ H if and only if M_{2}, M_{2}[v] ⊧ H. Let us observe that the predicate introduced above is reflexive.

The set Ax_{ZF} is defined by: (Def.4) e ∈ Ax_{ZF} if and only if e ∈ WFF but e = the axiom of extensionality or

e = the axiom of pairs or e = the axiom of unions or e = the axiom of infinity or e = the axiom of power sets or there exists H such that {x_{0}, x_{1}, x_{2}} misses Free H and e = the axiom of substitution for H.

Let us note that it makes sense to consider the following constant. Then Ax_{ZF} is a subset of WFF.

Let D be a non-empty set. Then ∅_{D} is a subset of D.

For simplicity we follow a convention: M, M_{1}, M_{2} will be non-empty families of sets, f will be a function, F, F_{1}, F_{2} will be subsets of WFF, W will be a universal class, a, b will be ordinals of W, A, B, C will be ordinal numbers, L will be a transfinite sequence of non-empty sets from W, and p_{1}, x_{1} will be transfinite sequences of ordinals of W. We now state a number of propositions:

(1) M ⊧ ∅_{WFF}.

(2) If F_{1} ⊆ F_{2} and M ⊧ F_{2}, then M ⊧ F_{1}.

(3) If M ⊧ F_{1} and M ⊧ F_{2}, then M ⊧ F_{1} ∪ F_{2}.

(4) If M is a model of ZF, then M ⊧ Ax_{ZF}.

(5) If M ⊧ Ax_{ZF} and M is transitive, then M is a model of ZF.

(6) There exists S such that Free S = ∅ and for every M holds M ⊧ S if and only if M ⊧ H.

(7) M_{1} ≡ M_{2} if and only if for every H holds M_{1} ⊧ H if and only if M_{2} ⊧ H.

(8) M_{1} ≡ M_{2} if and only if for every F holds M_{1} ⊧ F if and only if M_{2} ⊧ F.

(9) If M_{1} ≺ M_{2}, then M_{1} ≡ M_{2}.

(10) If M_{1} is a model of ZF and M_{1} ≡ M_{2} and M_{2} is transitive, then M_{2} is

a model of ZF. In this article we present several logical schemes. The scheme NonUniqBound-

Func deals with a set A, a set B, and a binary predicate P, and states that:

there exists a function f such that dom f = A and rng f ⊆ B and for every e such that e ∈ A holds P[e, f(e)] provided the following requirement is met:

• for every e such that e ∈ A there exists u such that u ∈ B and P[e, u].

The scheme NonUniqFuncEx deals with a set A, and a binary predicate P, and states that:

there exists a function f such that dom f = A and for every e such that e ∈ A holds P[e, f(e)] provided the following condition is met:

• for every e such that e ∈ A there exists u such that P[e, u].

The following propositions are true:

(11) If X ⊆ W and \overline{\overline{X}} < \overline{\overline{W}}, then X ∈ W.

(12) If dom f ∈ W and rng f ⊆ W, then rng f ∈ W.

(13) If X ≈ Y or \overline{\overline{X}} = \overline{\overline{Y}} , then 2^{X} ≈ 2^{Y} and \overline{\overline{2^{X}}} = \overline{\overline{2^{Y}}} .

(14) Let D be a non-empty set. Let P_{1} be a function from D into (On W)^{On W}. Suppose \overline{\overline{D}} < \overline{\overline{W}} and for every x_{1} such that x_{1} ∈ rng P_{1} holds x_{1} is increasing and x_{1} is continuous. Then there exists p_{1} such that p_{1} is increasing and p_{1} is continuous and p_{1}(0_{W}) = 0_{W} and for every a holds p_{1}(succ a) = sup({p_{1}(a)}∪uncurry P_{1}^{∘}[: D, {succ a}:]) and for every a such that a ≠ 0_{W} and a is a limit ordinal number holds p_{1}(a) = sup(p_{1} a).

(15) For every sequence p_{1} of ordinal numbers such that p_{1} is increasing holds C + p_{1} is increasing.

(16) For every sequence x_{1} of ordinal numbers holds (C +x_{1}) A = C +x_{1} A.

(17) For every sequence p_{1} of ordinal numbers such that p_{1} is increasing and

p_{1} is continuous holds C + p_{1} is continuous. Let A, B be ordinal numbers. We say that A is cofinal with B if and only if:

(Def.5) there exists a sequence x_{1} of ordinal numbers such that dom x_{1} = B

and rng x_{1} ⊆ A and x_{1} is increasing and A = sup x_{1}. Let us notice that the predicate defined above is reflexive.

In the sequel p_{2} will be a sequence of ordinal numbers. We now state a number of propositions:

(18) If p_{2} is increasing and A ⊆ B and B ∈ dom p_{2}, then p_{2}(A) ⊆ p_{2}(B).

(19) If e ∈ rng p_{2}, then e is an ordinal number.

(20) rng p_{2} ⊆ sup p_{2}.

(21) If A is cofinal with B and B is cofinal with C, then A is cofinal with C.

(22) If A is cofinal with B, then B ⊆ A.

(23) If A is cofinal with B and B is cofinal with A, then A = B.

(24) If dom p_{2} ≠ 0 and dom p_{2} is a limit ordinal number and p_{2} is increasing and A is the limit of p_{2}, then A is cofinal with dom p_{2}.

(25) succ A is cofinal with 1.

(26) If A is cofinal with succ B, then there exists C such that A = succ C.

(27) If A is cofinal with B, then A is a limit ordinal number if and only if B is a limit ordinal number.

(28) If A is cofinal with 0, then A = 0.

(29) On W is not cofinal with a.

(30) If ω ∈ W and p_{1} is increasing and p_{1} is continuous, then there exists b such that a ∈ b and p_{1}(b) = b.

(31) If ω ∈ W and p_{1} is increasing and p_{1} is continuous, then there exists a such that b ∈ a and p_{1}(a) = a and a is cofinal with ω.

(32) Suppose ω ∈ W and for all a, b such that a ∈ b holds L(a) ⊆ L(b) and for every a such that a ≠ 0 and a is a limit ordinal number holds L(a) = ⋃(L a). Then there exists p_{1} such that p_{1} is increasing and p_{1} is continuous and for every a such that p_{1}(a) = a and 0 ≠ a holds L(a) ≺ ⋃ L.

(33) R_{a} ∈ W.

(34) If a ≠ 0, then R_{a} is a non-empty set from W.

(35) If ω ∈ W, then there exists p_{1} such that p_{1} is increasing and p_{1} is continuous and for all a, M such that p_{1}(a) = a and 0 ≠ a and M = R_{a} holds M ≺ W.

(36) If ω ∈ W, then there exist b, M such that a ∈ b and M = R_{b} and M ≺ W.

(37) If ω ∈ W, then there exist a, M such that a is cofinal with ω and M = R_{a} and M ≺ W.

(38) Suppose ω ∈ W and for all a, b such that a ∈ b holds L(a) ⊆ L(b) and for every a such that a ≠ 0 and a is a limit ordinal number holds L(a) = ⋃(L a). Then there exists p_{1} such that p_{1} is increasing and p_{1} is continuous and for every a such that p_{1}(a) = a and 0 ≠ a holds L(a) ≡ ⋃ L.

(39) If ω ∈ W, then there exists p_{1} such that p_{1} is increasing and p_{1} is continuous and for all a, M such that p_{1}(a) = a and 0 ≠ a and M = R_{a} holds M ≡ W.

(40) If ω ∈ W, then there exist b, M such that a ∈ b and M = R_{b} and M ≡ W.

(41) If ω ∈ W, then there exist a, M such that a is cofinal with ω and M = R_{a} and M ≡ W.

(42) If ω ∈ W, then there exist a, M such that a is cofinal with ω and M = R_{a} and M is a model of ZF.

(43) If ω ∈ W and X ∈ W, then there exists M such that X ∈ M and M ∈ W and M is a model of ZF.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 1990.

[3] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711–714, 1990.

[4] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589– 593, 1990.

[5] Grzegorz Bancerek. A model of ZF set theory language. Formalized Mathematics, 1(1):131–145, 1990.

[6] Grzegorz Bancerek. Models and satisfiability. Formalized Mathematics, 1(1):191–199, 1990.

[7] Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515–519, 1990.

[8] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[9] Grzegorz Bancerek. The reflection theorem. Formalized Mathematics, 1(5):973–977, 1990.

[10] Grzegorz Bancerek. Replacing of variables in formulas of ZF theory. Formalized Mathematics, 1(5):963–972, 1990.

[11] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281–290, 1990.

[12] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.

[13] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[14] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[15] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[16] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

[17] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

Index of MML Identifiers

ALGSTR 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 ANALMETR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 ANPROJ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 ANPROJ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909 ANPROJ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 ANPROJ 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 ANPROJ 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939 FILTER 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 FINSOP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979 GROUP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 GROUP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 GROUP 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 INT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .829 LATTICE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 NET 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .949 NEWTON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 RAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .841 REALSET2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 RLVECT 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 VECTSP 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 VECTSP 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 VECTSP 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 VECTSP 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 VECTSP 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 ZFREFLE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989 ZF LANG1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963 ZF REFLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

Contents Form. Math. 1 (5)

Properties of Fields

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

Filters - Part I

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

Groups

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

The Divisibility of Integers and Integer Relatively Primes

By Rafał Kwiatek and Grzegorz Zwara . . . . . . . . . . . . . . . . . 829

From Loops to Abelian Multiplicative Groups with Zero

By Michał Muzalewski and Wojciech Skaba . . . . . . . . . . . . 833

Basic Properties of Rational Numbers

By Andrzej Kondracki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

Basis of Real Linear Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

Finite Sums of Vectors in Vector Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851

Subgroup and Cosets of Subgroups

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

Subspaces and Cosets of Subspaces in Vector Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865

Operations on Subspaces in Vector Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871

Linear Combinations in Vector Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877

Continued on inside back cover

Basis of Vector Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883

Factorial and Newton coeffitients

By Rafał Kwiatek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887

Analytical Metric Affine Spaces and Planes

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891

Projective Spaces - part II

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901

Projective Spaces - part III

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909

Projective Spaces - part IV

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919

Projective Spaces - part V

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929

Projective Spaces - part VI

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939

Some Elementary Notions of the Theory of Petri Nets

By Waldemar Korczyński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

Classes of Conjugation. Normal Subgroups

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

Replacing of Variables in Formulas of ZF Theory

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963

The Reflection Theorem

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

Binary Operations on Finite Sequences

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979

Finite Join and Finite Meet, and Dual Lattices

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983

Consequences of the Reflection Theorem

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994