Copyright (c) 1991 Association of Mizar Users
environ
vocabulary FUNCSDOM, VECTSP_1, ARYTM_1, RLVECT_1, VECTSP_2, RLVECT_2,
FINSEQ_1, FINSET_1, FUNCT_1, RELAT_1, SEQ_1, BOOLE, RLVECT_3, RLSUB_1,
FUNCT_2, PRELAMB, ZFMISC_1, TARSKI, ORDERS_1, MOD_3, HAHNBAN, FINSEQ_4;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NAT_1, FINSET_1, FINSEQ_1,
RELAT_1, FUNCT_1, STRUCT_0, FUNCT_2, FRAENKEL, FINSEQ_4, RLVECT_1,
ORDINAL1, ORDERS_1, RLVECT_2, VECTSP_1, FUNCSDOM, VECTSP_2, VECTSP_4,
VECTSP_5, VECTSP_6, LMOD_5;
constructors ORDERS_1, VECTSP_5, VECTSP_6, LMOD_5, RLVECT_2, FINSEQ_4,
MEMBERED, XBOOLE_0;
clusters VECTSP_2, VECTSP_4, RELSET_1, STRUCT_0, RLVECT_2, SUBSET_1, ARYTM_3,
VECTSP_1, MEMBERED, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;
definitions XBOOLE_0, TARSKI, LMOD_5, VECTSP_4, VECTSP_6;
theorems FINSEQ_1, FINSEQ_3, FINSEQ_4, FINSET_1, FUNCT_1, LMOD_5, MOD_1,
ORDERS_1, ORDERS_2, RLVECT_3, SUBSET_1, TARSKI, VECTSP_1, VECTSP_2,
ZFMISC_1, RLVECT_1, VECTSP_4, VECTSP_5, VECTSP_6, VECTSP_3, FUNCT_2,
RELAT_1, ORDINAL1, XBOOLE_0, XBOOLE_1, RLSUB_2;
schemes FUNCT_1, RLVECT_2, FUNCT_2, XBOOLE_0;
begin
Lm1: for R being Ring, a being Scalar of R holds -a = 0.R implies a = 0.R
proof
let R be Ring, a be Scalar of R;
assume -a = 0.R;
then --a = 0.R by RLVECT_1:25;
hence thesis by RLVECT_1:30;
end;
canceled;
theorem Th2:
for R being non degenerated add-associative right_zeroed
right_complementable (non empty doubleLoopStr) holds 0.R <> -1_ R
proof
let R be non degenerated add-associative right_zeroed
right_complementable (non empty doubleLoopStr);
assume 0.R = -1_ R;
then 0.R = -(-1_ R) by RLVECT_1:25
.= 1_ R by RLVECT_1:30;
hence contradiction by VECTSP_1:def 21;
end;
reserve x,y for set,
R for Ring,
V for LeftMod of R,
L for Linear_Combination of V,
a for Scalar of R,
v,u for Vector of V,
F,G for FinSequence of the carrier of V,
C for finite Subset of V;
canceled 3;
theorem
Th6: Carrier(L) c= C implies ex F st
F is one-to-one & rng F = C & Sum(L) = Sum(L (#) F)
proof
assume
A1: Carrier(L) c= C;
consider G such that
A2: G is one-to-one and
A3: rng G = Carrier(L) and
A4: Sum(L) = Sum(L (#) G) by VECTSP_6:def 9;
set D = C \ Carrier(L);
consider p being FinSequence such that
A5: rng p = D and
A6: p is one-to-one by FINSEQ_4:73;
reconsider p as FinSequence of the carrier of V by A5,FINSEQ_1:def 4;
A7: len p = len(L (#) p) by VECTSP_6:def 8;
now
let k be Nat;
assume
A8: k in dom p;
then k in dom(L (#) p) by A7,FINSEQ_3:31;
then A9: (L (#) p).k = L.(p/.k) * (p/.k) by VECTSP_6:def 8;
p/.k = p.k by A8,FINSEQ_4:def 4;
then p/.k in D by A5,A8,FUNCT_1:def 5;
then not p/.k in Carrier(L) by XBOOLE_0:def 4;
hence (L (#) p).k = 0.R * (p/.k) by A9,VECTSP_6:20;end;
then A10: Sum(L (#) p) = 0.R * Sum(p) by A7,VECTSP_3:10
.= 0.V by VECTSP_1:59;
set F = G ^ p;
A11: Sum((L (#) F)) = Sum((L (#) G) ^ (L (#) p)) by VECTSP_6:37
.= Sum(L (#) G) + 0.V by A10,RLVECT_1:58
.= Sum(L) by A4,VECTSP_1:7;
A12: rng G misses rng p
proof
assume rng G meets rng p;
then consider x being set such that
A13: x in Carrier(L) & x in D by A3,A5,XBOOLE_0:3;
thus thesis by A13,XBOOLE_0:def 4;
end;
A14: rng F = Carrier(L) \/ D by A3,A5,FINSEQ_1:44
.= C by A1,XBOOLE_1:45;
take F;
thus thesis by A2,A6,A11,A12,A14,FINSEQ_3:98;
end;
theorem
Th7: Sum(a * L) = a * Sum(L)
proof
set l = a * L;
Carrier(l) c= Carrier(L) by VECTSP_6:58;
then consider F such that
A1: F is one-to-one and
A2: rng F = Carrier(L) and
A3: Sum(l) = Sum(l (#) F) by Th6;
A4: Sum(L) = Sum(L (#) F) by A1,A2,VECTSP_6:def 9;
set f = l (#) F, g = L (#) F;
A5: len f = len F by VECTSP_6:def 8
.= len g by VECTSP_6:def 8;
len f = len F by VECTSP_6:def 8;
then A6: dom F = Seg len f by FINSEQ_1:def 3;
now
let k be Nat, v; assume
A7: k in dom f & v = g.k;
then A8: k in Seg len f by FINSEQ_1:def 3;
set v' = F/.k;
A9: v' = F.k by A6,A8,FINSEQ_4:def 4;
hence f.k = l.v'*v' by A6,A8,VECTSP_6:32
.= (a*L.v')*v' by VECTSP_6:def 12
.= a*(L.v'*v') by VECTSP_1:def 26
.= a*v by A6,A7,A8,A9,VECTSP_6:32;
end;
hence thesis by A3,A4,A5,VECTSP_3:9;
end;
reserve X,Y,Z for set,
A,B for Subset of V,
T for finite Subset of V,
l for Linear_Combination of A,
f,g for Function of the carrier of V,the carrier of R;
definition let R,V,A;
func Lin(A) -> strict Subspace of V means :Def1:
the carrier of it = {Sum(l) : not contradiction};
existence
proof
set A1 = {Sum(l) : not contradiction};
A1 c= the carrier of V
proof
let x;
assume x in A1;
then ex l st x = Sum(l);
hence thesis;
end;
then reconsider A1 as Subset of V;
reconsider l = ZeroLC(V) as Linear_Combination of A by VECTSP_6:26;
Sum(l) = 0.V by VECTSP_6:41;
then A1: 0.V in A1;
A1 is lineary-closed
proof
thus for v,u st v in A1 & u in A1 holds v + u in A1
proof
let v,u;
assume that
A2: v in A1 and
A3: u in A1;
consider l1 being Linear_Combination of A such that
A4: v = Sum(l1) by A2;
consider l2 being Linear_Combination of A such that
A5: u = Sum(l2) by A3;
reconsider f = l1 + l2 as Linear_Combination of A by VECTSP_6:52;
v + u = Sum(f) by A4,A5,VECTSP_6:77;
hence thesis;
end;
let a,v;
assume v in A1;
then consider l such that
A6: v = Sum(l);
reconsider f = a * l as Linear_Combination of A by VECTSP_6:61;
a * v = Sum(f) by A6,Th7;
hence thesis;
end;
hence thesis by A1,VECTSP_4:42;
end;
uniqueness by VECTSP_4:37;
end;
canceled 3;
theorem Th11:
x in Lin(A) iff ex l st x = Sum(l)
proof
thus x in Lin(A) implies ex l st x = Sum(l)
proof assume x in Lin(A);
then x in the carrier of Lin(A) by RLVECT_1:def 1;
then x in {Sum(l) : not contradiction} by Def1;
hence thesis;
end;
given k being Linear_Combination of A such that A1: x = Sum(k);
x in {Sum(l): not contradiction} by A1;
then x in the carrier of Lin(A) by Def1;
hence thesis by RLVECT_1:def 1;
end;
theorem Th12:
x in A implies x in Lin(A)
proof assume A1: x in A;
then reconsider v = x as Vector of V;
deffunc F(Vector of V) = 0.R;
consider f being
Function of the carrier of V, the carrier of R such that
A2: f.v = 1_ R and
A3: for u st u <> v holds f.u = F(u) from LambdaSep1;
reconsider f as
Element of Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
ex T st for u st not u in T holds f.u = 0.R
proof take T = {v};
let u;
assume not u in T;
then u <> v by TARSKI:def 1;
hence thesis by A3;
end;
then reconsider f as Linear_Combination of V by VECTSP_6:def 4;
A4: Carrier(f) c= {v}
proof let x;
assume x in Carrier(f);
then x in {u : f.u <> 0.R} by VECTSP_6:def 5;
then consider u such that A5: x = u and A6: f.u <> 0.R;
u = v by A3,A6;
hence thesis by A5,TARSKI:def 1;
end;
then reconsider f as Linear_Combination of {v} by VECTSP_6:def 7;
A7: Sum(f) = 1_ R * v by A2,VECTSP_6:43
.= v by VECTSP_1:def 26;
{v} c= A by A1,ZFMISC_1:37;
then Carrier(f) c= A by A4,XBOOLE_1:1;
then reconsider f as Linear_Combination of A by VECTSP_6:def 7;
Sum(f) = v by A7;
hence thesis by Th11;
end;
theorem
Th13: Lin({}(the carrier of V)) = (0).V
proof
set A = Lin({}(the carrier of V));
now
let v;
thus v in A implies v in (0).V
proof
assume v in A;
then v in the carrier of A & the carrier of A =
{Sum(l0) where l0 is Linear_Combination of
{}(the carrier of V): not contradiction} by Def1,RLVECT_1:def 1;
then ex l0 being Linear_Combination of
{}(the carrier of V) st v = Sum(l0);
then v = 0.V by VECTSP_6:42;
hence thesis by VECTSP_4:46;
end;
assume v in (0).V;
then v = 0.V by VECTSP_4:46;
hence v in A by VECTSP_4:25;end;
hence thesis by VECTSP_4:38;
end;
theorem
Lin(A) = (0).V implies A = {} or A = {0.V}
proof assume that A1: Lin(A) = (0).V and A2: A <> {};
thus A c= {0.V}
proof let x;
assume x in A;
then x in Lin(A) by Th12;
then x = 0.V by A1,VECTSP_4:46;
hence thesis by TARSKI:def 1;
end;
let x;
assume x in {0.V};
then A3: x = 0.V by TARSKI:def 1;
consider y being Element of A;
A4: y in A by A2;
y in Lin(A) by A2,Th12;
hence thesis by A1,A3,A4,VECTSP_4:46;
end;
theorem Th15:
for W being strict Subspace of V
holds 0.R <> 1_ R & A = the carrier of W implies Lin(A) = W
proof let W be strict Subspace of V;
assume that A1: 0.R <> 1_ R and
A2: A = the carrier of W;
now let v;
thus v in Lin(A) implies v in W
proof assume v in Lin(A);
then A3: ex l st v = Sum(l) by Th11;
A is lineary-closed & A <> {} by A2,VECTSP_4:41;
then v in the carrier of W by A1,A2,A3,VECTSP_6:40;
hence thesis by RLVECT_1:def 1;
end;
v in W iff v in the carrier of W by RLVECT_1:def 1;
hence v in W implies v in Lin(A) by A2,Th12;
end;
hence thesis by VECTSP_4:38;
end;
theorem
for V being strict LeftMod of R
for A being Subset of V
holds 0.R <> 1_ R & A = the carrier of V implies Lin(A) = V
proof let V be strict LeftMod of R;
let A be Subset of V;
assume that A1: 0.R <> 1_ R and
A2: A = the carrier of V;
reconsider B = (Omega).V as Subspace of V;
A = the carrier of (Omega).V by A2,VECTSP_4:def 4;
hence Lin(A) = B by A1,Th15
.= V by VECTSP_4:def 4;
end;
theorem Th17:
A c= B implies Lin(A) is Subspace of Lin(B)
proof assume A1: A c= B;
now let v;
assume v in Lin(A);
then consider l such that A2: v = Sum(l) by Th11;
reconsider l as Linear_Combination of B by A1,VECTSP_6:25;
Sum(l) = v by A2;
hence v in Lin(B) by Th11;
end;
hence thesis by VECTSP_4:36;
end;
theorem
Lin(A) = V & A c= B implies Lin(B) = V
proof
assume
A1: Lin(A) = V & A c= B;
then V is Subspace of Lin(B) by Th17;
hence thesis by A1,VECTSP_4:33;
end;
theorem
Lin(A \/ B) = Lin(A) + Lin(B)
proof A c= A \/ B & B c= A \/ B by XBOOLE_1:7;
then Lin(A) is Subspace of Lin(A \/ B) &
Lin(B) is Subspace of Lin(A \/ B) by Th17;
then A1: Lin(A) + Lin(B) is Subspace of Lin(A \/ B) by VECTSP_5:40;
now let v;
assume v in Lin(A \/ B);
then consider l being Linear_Combination of A \/ B such that
A2: v = Sum(l) by Th11;
set C = Carrier(l) /\ A; set D = Carrier(l) \ A;
deffunc F(set)=l.$1;
deffunc G(set)=0.R;
defpred C[set] means $1 in C;
A3: for x st x in the carrier of V holds
(C[x] implies F(x) in the carrier of R) &
(not C[x] implies G(x) in the carrier of R)
proof let x;
assume x in the carrier of V;
then reconsider v = x as Vector of V;
f.v in the carrier of R;
hence x in C implies l.x in the carrier of R;
assume not x in C;
thus 0.R in the carrier of R;
end;
consider f being
Function of the carrier of V, the carrier of R such that
A4: for x st x in the carrier of V holds
(C[x] implies f.x = F(x)) & (not C[x] implies f.x = G(x))
from Lambda1C(A3);
reconsider f as
Element of Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
defpred D[set] means $1 in D;
A5:for x st x in the carrier of V holds
(D[x] implies F(x) in the carrier of R) &
(not D[x] implies G(x) in the carrier of R)
proof let x;
assume x in the carrier of V;
then reconsider v = x as Vector of V;
g.v in the carrier of R;
hence x in D implies l.x in the carrier of R;
assume not x in D;
thus 0.R in the carrier of R;
end;
consider g being
Function of the carrier of V, the carrier of R such that
A6: for x st x in the carrier of V holds
(D[x] implies g.x = F(x)) & (not D[x] implies g.x = G(x))
from Lambda1C(A5);
reconsider g as
Element of Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
C c= Carrier(l) & Carrier(l) is finite by XBOOLE_1:17;
then reconsider C as finite Subset of V by FINSET_1:13;
for u holds not u in C implies f.u = 0.R by A4;
then reconsider f as Linear_Combination of V by VECTSP_6:def 4;
A7: Carrier(f) c= C
proof let x;
assume x in Carrier(f);
then x in {u : f.u <> 0.R} by VECTSP_6:def 5;
then A8: ex u st x = u & f.u <> 0.R;
assume not x in C;
hence thesis by A4,A8;
end;
C c= A by XBOOLE_1:17;
then Carrier(f) c= A by A7,XBOOLE_1:1;
then reconsider f as Linear_Combination of A by VECTSP_6:def 7;
D c= Carrier(l) & Carrier(l) is finite by XBOOLE_1:36;
then reconsider D as finite Subset of V by FINSET_1:13;
for u holds not u in D implies g.u = 0.R by A6;
then reconsider g as Linear_Combination of V by VECTSP_6:def 4;
A9: Carrier(g) c= D
proof let x;
assume x in Carrier(g);
then x in {u : g.u <> 0.R} by VECTSP_6:def 5;
then A10: ex u st x = u & g.u <> 0.R;
assume not x in D;
hence thesis by A6,A10;
end;
D c= B
proof let x;
assume x in D;
then x in Carrier(l) & not x in A & Carrier(l) c= A \/ B
by VECTSP_6:def 7,XBOOLE_0:def 4;
hence thesis by XBOOLE_0:def 2;
end;
then Carrier(g) c= B by A9,XBOOLE_1:1;
then reconsider g as Linear_Combination of B by VECTSP_6:def 7;
l = f + g
proof let v;
now per cases;
suppose A11: v in C;
A12: now assume v in D;
then not v in A by XBOOLE_0:def 4;
hence contradiction by A11,XBOOLE_0:def 3;
end;
thus (f + g).v = f.v + g.v by VECTSP_6:def 11
.= l.v + g.v by A4,A11
.= l.v + 0.R by A6,A12
.= l.v by RLVECT_1:10;
suppose A13: not v in C;
now per cases;
suppose A14: v in Carrier(l);
A15: now assume not v in D;
then not v in Carrier(l) or v in A by XBOOLE_0:def 4;
hence contradiction by A13,A14,XBOOLE_0:def 3;
end;
thus (f + g). v = f.v + g.v by VECTSP_6:def 11
.= 0.R + g.v by A4,A13
.= g.v by RLVECT_1:10
.= l.v by A6,A15;
suppose A16: not v in Carrier(l);
then A17: not v in C & not v in D by XBOOLE_0:def 3,def 4;
thus (f + g).v = f.v + g.v by VECTSP_6:def 11
.= 0.R + g.v by A4,A17
.= 0.R + 0.R by A6,A17
.= 0.R by RLVECT_1:10
.= l.v by A16,VECTSP_6:20;
end;
hence (f + g).v = l.v;
end;
hence thesis;
end;
then A18: v = Sum(f) + Sum(g) by A2,VECTSP_6:77;
Sum(f) in Lin(A) & Sum(g) in Lin(B) by Th11;
hence v in Lin(A) + Lin(B) by A18,VECTSP_5:5;
end;
then Lin(A \/ B) is Subspace of Lin(A) + Lin(B) by VECTSP_4:36;
hence thesis by A1,VECTSP_4:33;
end;
theorem
Lin(A /\ B) is Subspace of Lin(A) /\ Lin(B)
proof A /\ B c= A & A /\ B c= B by XBOOLE_1:17;
then Lin(A /\ B) is Subspace of Lin(A) & Lin(A /\
B) is Subspace of Lin(B) by Th17;
hence thesis by VECTSP_5:24;
end;
definition let R,V;
let IT be Subset of V;
attr IT is base means
:Def2: IT is linearly-independent & Lin(IT) = the VectSpStr of V;
end;
definition let R;
let IT be LeftMod of R;
attr IT is free means
:Def3: ex B being Subset of IT st B is base;
end;
theorem
Th21: (0).V is free
proof
set W = (0).V;
reconsider B' = {}(the carrier of V)
as Subset of W by SUBSET_1:4;
A1: B' = {}(the carrier of W);
then A2: B' is linearly-independent by LMOD_5:4;
reconsider V' = V as Subspace of V by VECTSP_4:32;
(0).V' = (0).W by VECTSP_4:48;
then Lin(B') = W by A1,Th13;
then B' is base by A2,Def2;
hence thesis by Def3;
end;
definition let R;
cluster strict free LeftMod of R;
existence
proof
((0).(LeftModule R)) is free by Th21;
hence thesis;
end;
end;
reserve R for Skew-Field;
reserve a,b for Scalar of R;
reserve V for LeftMod of R;
reserve v,v1,v2,u for Vector of V;
reserve f for Function of the carrier of V,the carrier of R;
Lm2: a <> 0.R implies a"*(a*v) = 1_ R*v & (a"*a)*v = 1_ R*v
proof
assume
A1: a <> 0.R;
hence a"*(a*v) = v by MOD_1:26
.= 1_ R*v by VECTSP_1:def 26;
thus (a"*a)*v = 1_ R*v by A1,VECTSP_2:43;
end;
canceled;
theorem
{v} is linearly-independent iff v <> 0.V
proof
A1: 0.R <> 1_ R by VECTSP_1:def 21;
thus {v} is linearly-independent implies v <> 0.V
proof
assume {v} is linearly-independent;
then not 0.V in {v} by A1,LMOD_5:3;
hence thesis by TARSKI:def 1;
end;
assume A2: v <> 0.V;
let l be Linear_Combination of {v};
assume A3: Sum(l) = 0.V;
A4: Carrier(l) c= {v} by VECTSP_6:def 7;
now per cases by A4,ZFMISC_1:39;
suppose Carrier(l) = {};
hence thesis;
suppose A5: Carrier(l) = {v};
A6: 0.V = l.v * v by A3,VECTSP_6:43;
now assume v in Carrier(l);
then l.v <> 0.R by VECTSP_6:20;
hence contradiction by A2,A6,MOD_1:25;
end;
hence thesis by A5,TARSKI:def 1;
end;
hence thesis;
end;
theorem Th24:
v1 <> v2 & {v1,v2} is linearly-independent iff
v2 <> 0.V & for a holds v1 <> a * v2
proof
A1: 0.R <> 1_ R by VECTSP_1:def 21;
thus v1 <> v2 & {v1,v2} is linearly-independent implies
v2 <> 0.V & for a holds v1 <> a * v2
proof assume that A2: v1 <> v2 and A3: {v1,v2} is linearly-independent;
thus v2 <> 0.V by A1,A3,LMOD_5:5;
let a;
assume A4: v1 = a * v2;
deffunc F(Element of V) = 0.R;
consider f such that A5: f.v1 = - 1_ R & f.v2 = a and
A6: for v being Element of V st
v <> v1 & v <> v2 holds f.v = F(v) from LambdaSep2(A2);
reconsider f as
Element of Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
now let v;
assume not v in ({v1,v2});
then v <> v1 & v <> v2 by TARSKI:def 2;
hence f.v = 0.R by A6;
end;
then reconsider f as Linear_Combination of V by VECTSP_6:def 4;
Carrier(f) c= {v1,v2}
proof let x;
assume x in Carrier(f);
then x in {u : f.u <> 0.R} by VECTSP_6:def 5;
then A7: ex u st x = u & f.u <> 0.R;
assume not x in {v1,v2};
then x <> v1 & x <> v2 by TARSKI:def 2;
hence thesis by A6,A7;
end;
then reconsider f as Linear_Combination of {v1,v2} by VECTSP_6:def 7;
A8: now assume not v1 in Carrier(f);
then not v1 in {u : f.u <> 0.R} by VECTSP_6:def 5;
then 0.R = - 1_ R by A5;
hence contradiction by Th2;
end;
set w = a * v2;
Sum(f) = (- 1_ R) * w + w by A2,A4,A5,VECTSP_6:44
.= (- w) + w by VECTSP_1:59
.= 0.V by RLVECT_1:16;
hence thesis by A3,A8,LMOD_5:def 1;
end;
assume A9: v2 <> 0.V;
assume A10: for a holds v1 <> a * v2;
then A11: v1 <> 1_ R * v2 & 1_ R * v2 = v2 by VECTSP_1:def 26;
hence v1 <> v2;
let l be Linear_Combination of {v1,v2};
assume that A12: Sum(l) = 0.V and A13: Carrier(l) <> {};
consider x being Element of Carrier(l);
x in Carrier(l) by A13;
then x in {u : l.u <> 0.R} by VECTSP_6:def 5;
then A14: ex u st x = u & l.u <> 0.R;
Carrier(l) c= {v1,v2} by VECTSP_6:def 7;
then A15: x in {v1,v2} by A13,TARSKI:def 3;
A16: 0.V = l.v1 * v1 + l.v2 * v2 by A11,A12,VECTSP_6:44;
now per cases by A14,A15,TARSKI:def 2;
suppose A17: l.v1 <> 0.R;
0.V = (l.v1)" * (l.v1 * v1 + l.v2 * v2) by A16,MOD_1:25
.= (l.v1)" * (l.v1 * v1) + (l.v1)" * (l.v2 * v2)
by VECTSP_1:def 26
.= (l.v1)" * l.v1 * v1 + (l.v1)" * (l.v2 * v2) by VECTSP_1:def 26
.= (l.v1)" * l.v1 * v1 + (l.v1)" * l.v2 * v2 by VECTSP_1:def 26
.= 1_ R * v1 + (l.v1)" * l.v2 * v2 by A17,Lm2
.= v1 + (l.v1)" * l.v2 * v2 by VECTSP_1:def 26;
then v1 = - ((l.v1)" * l.v2 * v2) by VECTSP_1:63
.= (- 1_ R) * ((l.v1)" * l.v2 * v2) by VECTSP_1:59
.= ((- 1_ R) * ((l.v1)" * l.v2)) * v2 by VECTSP_1:def 26;
hence thesis by A10;
suppose A18: l.v2 <> 0.R & l.v1 = 0.R;
0.V = (l.v2)" * (l.v1 * v1 + l.v2 * v2) by A16,MOD_1:25
.= (l.v2)" * (l.v1 * v1) + (l.v2)" * (l.v2 * v2)
by VECTSP_1:def 26
.= (l.v2)" * l.v1 * v1 + (l.v2)" * (l.v2 * v2) by VECTSP_1:def 26
.= (l.v2)" * l.v1 * v1 + 1_ R * v2 by A18,Lm2
.= (l.v2)" * l.v1 * v1 + v2 by VECTSP_1:def 26
.= 0.R * v1 + v2 by A18,VECTSP_1:36
.= 0.V + v2 by MOD_1:25
.= v2 by VECTSP_1:7;
hence thesis by A9;
end;
hence thesis;
end;
theorem
v1 <> v2 & {v1,v2} is linearly-independent iff
for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R
proof
thus v1 <> v2 & {v1,v2} is linearly-independent implies
for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R
proof assume that A1: v1 <> v2 and A2: {v1,v2} is linearly-independent;
let a,b;
assume that A3: a * v1 + b * v2 = 0.V and A4: a <> 0.R or b <> 0.R;
now per cases by A4;
suppose A5: a <> 0.R;
0.V = a" * (a * v1 + b * v2) by A3,MOD_1:25
.= a" * (a * v1) + a" * (b * v2) by VECTSP_1:def 26
.= (a" * a) * v1 + a" * (b * v2) by VECTSP_1:def 26
.= (a" * a) * v1 + (a" * b) * v2 by VECTSP_1:def 26
.= 1_ R * v1 + (a" * b) * v2 by A5,Lm2
.= v1 + (a" * b) * v2 by VECTSP_1:def 26;
then v1 = - ((a" * b) * v2) by VECTSP_1:63
.= (- 1_ R) * ((a" * b) * v2) by VECTSP_1:59
.= (- 1_ R) * (a" * b) * v2 by VECTSP_1:def 26;
hence thesis by A1,A2,Th24;
suppose A6: b <> 0.R;
0.V = b" * (a * v1 + b * v2) by A3,MOD_1:25
.= b" * (a * v1) + b" * (b * v2) by VECTSP_1:def 26
.= (b" * a) * v1 + b" * (b * v2) by VECTSP_1:def 26
.= (b" * a) * v1 + 1_ R* v2 by A6,Lm2
.= (b" * a) * v1 + v2 by VECTSP_1:def 26;
then v2 = - ((b" * a) * v1) by VECTSP_1:63
.= (- 1_ R) * ((b" * a) * v1) by VECTSP_1:59
.= (- 1_ R) * (b" * a) * v1 by VECTSP_1:def 26;
hence thesis by A1,A2,Th24;
end;
hence thesis;
end;
assume A7: for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R;
A8: now assume A9: v2 = 0.V;
0.V = 0.V + 0.V by VECTSP_1:7
.= 0.R * v1 + 0.V by MOD_1:25
.= 0.R * v1 + 1_ R * v2 by A9,MOD_1:25;
then 0.R = 1_ R by A7;
hence contradiction by VECTSP_1:def 21;
end;
now let a;
assume v1 = a * v2;
then v1 = 0.V + a * v2 by VECTSP_1:7;
then 0.V = v1 - a * v2 by RLSUB_2:78
.= v1 + ((- a) * v2) by VECTSP_1:68
.= 1_ R * v1 + (- a) * v2 by VECTSP_1:def 26;
then 1_ R = 0.R by A7;
hence contradiction by VECTSP_1:def 21;
end;
hence thesis by A8,Th24;
end;
theorem
Th26: for V being LeftMod of R
for A being Subset of V
holds A is linearly-independent implies
ex B being Subset of V st A c= B & B is base
proof let V be LeftMod of R;
let A be Subset of V;
assume A1: A is linearly-independent;
defpred P[set] means (ex B being Subset of V
st B = $1 & A c= B & B is linearly-independent);
consider Q being set such that
A2: for Z holds Z in Q iff Z in bool(the carrier of V) &
P[Z] from Separation;
A3: Q <> {} by A1,A2;
now let Z;
assume that A4: Z <> {} and A5: Z c= Q and
A6: Z is c=-linear;
set W = union Z;
W c= the carrier of V
proof let x;
assume x in W;
then consider X such that A7: x in X and A8: X in Z by TARSKI:def
4;
X in bool(the carrier of V) by A2,A5,A8;
hence thesis by A7;
end;
then reconsider W as Subset of V;
consider x being Element of Z;
x in Q by A4,A5,TARSKI:def 3;
then A9: ex B being Subset of V st B = x & A c= B &
B is linearly-independent by A2;
x c= W by A4,ZFMISC_1:92;
then A10: A c= W by A9,XBOOLE_1:1;
W is linearly-independent
proof let l be Linear_Combination of W;
assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {};
deffunc F(set)={C where C is Subset of V :
$1 in C & C in Z};
consider f being Function such that
A13: dom f = Carrier(l) and
A14: for x st x in Carrier(l) holds
f.x = F(x) from Lambda;
reconsider M = rng f as non empty set by A12,A13,RELAT_1:65;
consider F being Choice_Function of M;
A15: now assume {} in M;
then consider x such that A16: x in dom f and A17: f.x = {}
by FUNCT_1:def 5;
Carrier(l) c= W by VECTSP_6:def 7;
then consider X such that A18: x in X and
A19: X in Z by A13,A16,TARSKI:def 4;
reconsider X as Subset of V by A2,A5,A19;
X in {C where C is Subset of V :
x in C & C in Z} by A18,A19;
hence contradiction by A13,A14,A16,A17;
end;
set S = rng F;
A20: dom F = M & M <> {} by A15,RLVECT_3:35;
then A21: S <> {} by RELAT_1:65;
A22: now let X;
assume X in S;
then consider x such that A23: x in dom F and A24: F.x = X
by FUNCT_1:def 5;
consider y such that A25: y in dom f and A26: f.y = x
by A20,A23,FUNCT_1:def 5;
A27: X in x by A15,A20,A23,A24,ORDERS_1:def 1;
x = {C where C is Subset of V:
y in C & C in Z} by A13,A14,A25,A26;
then ex C being Subset of V st
C = X & y in C & C in Z by A27;
hence X in Z;
end;
A28: now let X,Y;
assume X in S & Y in S;
then X in Z & Y in Z by A22;
then X,Y are_c=-comparable by A6,ORDINAL1:def 9;
hence X c= Y or Y c= X by XBOOLE_0:def 9;
end;
dom F is finite by A13,A20,FINSET_1:26;
then S is finite by FINSET_1:26;
then union S in S by A21,A28,RLVECT_3:34;
then union S in Z by A22;
then consider B being Subset of V such that
A29: B = union S and A c= B and
A30: B is linearly-independent by A2,A5;
Carrier(l) c= union S
proof let x;
assume A31: x in Carrier(l);
then A32: f.x in M by A13,FUNCT_1:def 5;
set X = f.x;
A33: F.X in X by A15,A32,ORDERS_1:def 1;
f.x = {C where C is Subset of V :
x in C & C in Z} by A14,A31;
then A34: ex C being Subset of V st
F.X = C & x in C & C in Z by A33;
F.X in S by A20,A32,FUNCT_1:def 5;
hence x in union S by A34,TARSKI:def 4;
end;
then l is Linear_Combination of B by A29,VECTSP_6:def 7;
hence thesis by A11,A12,A30,LMOD_5:def 1;
end;
hence union Z in Q by A2,A10;
end;
then consider X such that A35: X in Q and
A36: for Z st Z in Q & Z <> X holds not X c= Z by A3,ORDERS_2:79;
consider B being Subset of V such that
A37: B = X and A38: A c= B and
A39: B is linearly-independent by A2,A35;
take B;
thus A c= B & B is linearly-independent by A38,A39;
A40: the VectSpStr of V = (Omega).V by VECTSP_4:def 4;
assume Lin(B) <> the VectSpStr of V;
then consider v being Vector of V such that
A41: not (v in Lin(B) iff v in (Omega).V) by A40,VECTSP_4:38;
A42: B \/ {v} is linearly-independent
proof let l be Linear_Combination of B \/ {v};
assume A43: Sum(l) = 0.V;
now per cases;
suppose A44: v in Carrier(l);
deffunc F(Vector of V) = l.$1;
consider f being
Function of the carrier of V,
the carrier of R such that A45: f.v = 0.R and
A46: for u being Vector of V st u <> v
holds f.u = F(u) from LambdaSep1;
reconsider f as
Element of Funcs(the carrier of V,
the carrier of R) by FUNCT_2:11;
now let u be Vector of V;
assume not u in Carrier(l) \ {v};
then not u in Carrier(l) or u in {v} by XBOOLE_0:def 4;
then (l.u = 0.R & u <> v) or u = v
by TARSKI:def 1,VECTSP_6:20;
hence f.u = 0.R by A45,A46;
end;
then reconsider f as Linear_Combination of V by VECTSP_6:def 4;
Carrier(f) c= B
proof let x;
assume x in Carrier(f);
then x in {u where u is Vector of V :
f.u <> 0.R} by VECTSP_6:def 5;
then consider u being Vector of V such that
A47: u = x and A48: f.u <> 0.R;
f.u = l.u by A45,A46,A48;
then u in {v1 where v1 is Vector of V :
l.v1 <> 0.R} by A48;
then u in Carrier(l) & Carrier(l) c= B \/ {v}
by VECTSP_6:def 5,def 7;
then u in B or u in {v} by XBOOLE_0:def 2;
hence thesis by A45,A47,A48,TARSKI:def 1;
end;
then reconsider f as Linear_Combination of B by VECTSP_6:def 7;
deffunc F(Vector of V)=0.R;
consider g being Function of the carrier of V,
the carrier of R such that A49: g.v = - l.v and
A50: for u being Vector of V st u <> v
holds g.u = F(u) from LambdaSep1;
reconsider g as Element of
Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
now let u be Vector of V;
assume not u in {v};
then u <> v by TARSKI:def 1;
hence g.u = 0.R by A50;
end;
then reconsider g as Linear_Combination of V by VECTSP_6:def 4;
Carrier(g) c= {v}
proof let x;
assume x in Carrier(g);
then x in {u where u is Vector of V :
g.u <> 0.R} by VECTSP_6:def 5;
then ex u being Vector of V st x = u & g.u <> 0.R;
then x = v by A50;
hence thesis by TARSKI:def 1;
end;
then reconsider g as Linear_Combination of {v} by VECTSP_6:def 7;
A51: f - g = l
proof let u be Vector of V;
now per cases;
suppose A52: v = u;
thus (f - g).u = f.u - g.u by VECTSP_6:73
.= 0.R + (- (- l.v))
by A45,A49,A52,RLVECT_1:def 11
.= l.v + 0.R by RLVECT_1:30
.= l.u by A52,RLVECT_1:10;
suppose A53: v <> u;
thus (f - g).u = f.u - g.u by VECTSP_6:73
.= l.u - g.u by A46,A53
.= l.u - 0.R by A50,A53
.= l.u by RLVECT_1:26;
end;
hence thesis;
end;
A54: Sum(g) = (- l.v) * v by A49,VECTSP_6:43;
0.V = Sum(f) - Sum(g) by A43,A51,VECTSP_6:80;
then Sum(f) = (- l.v) * v by A54,VECTSP_1:66;
then A55: (- l.v) * v in Lin(B) by Th11;
l.v <> 0.R by A44,VECTSP_6:20;
then - l.v <> 0.R by Lm1;
then (- l.v)" * ((- l.v) * v) = 1_ R * v by Lm2
.= v by VECTSP_1:def 26;
hence thesis by A40,A41,A55,RLVECT_1:def 1,VECTSP_4:29;
suppose A56: not v in Carrier(l);
Carrier(l) c= B
proof let x;
assume A57: x in Carrier(l);
Carrier(l) c= B \/ {v} by VECTSP_6:def 7;
then x in B or x in {v} by A57,XBOOLE_0:def 2;
hence thesis by A56,A57,TARSKI:def 1;
end;
then l is Linear_Combination of B by VECTSP_6:def 7;
hence thesis by A39,A43,LMOD_5:def 1;
end;
hence thesis;
end;
v in {v} by TARSKI:def 1;
then A58: v in B \/ {v} & not v in B
by A40,A41,Th12,RLVECT_1:def 1,XBOOLE_0:def 2;
A59: B c= B \/ {v} by XBOOLE_1:7;
then A c= B \/ {v} by A38,XBOOLE_1:1;
then B \/ {v} in Q by A2,A42;
hence contradiction by A36,A37,A58,A59;
end;
theorem Th27:
for V being LeftMod of R
for A being Subset of V
holds Lin(A) = V implies ex B being Subset of V st
B c= A & B is base
proof let V be LeftMod of R;
let A be Subset of V;
A1: 0.R <> 1_ R by VECTSP_1:def 21;
assume A2: Lin(A) = V;
defpred P[set] means (ex B being Subset of V
st B = $1 & B c= A & B is linearly-independent);
consider Q being set such that
A3: for Z holds Z in Q iff Z in bool(the carrier of V) &
P[Z] from Separation;
{}(the carrier of V) in
bool(the carrier of V) &
{}(the carrier of V) c= A &
{}(the carrier of V) is linearly-independent by LMOD_5:4,XBOOLE_1:2;
then A4: Q <> {} by A3;
now let Z;
assume that Z <> {} and A5: Z c= Q and
A6: Z is c=-linear;
set W = union Z;
W c= the carrier of V
proof let x;
assume x in W;
then consider X such that A7: x in X and A8: X in Z by TARSKI:def
4;
X in bool(the carrier of V) by A3,A5,A8;
hence thesis by A7;
end;
then reconsider W as Subset of V;
A9: W c= A
proof let x;
assume x in W;
then consider X such that A10: x in X and A11: X in Z by TARSKI:def
4;
ex B being Subset of V st B = X & B c= A &
B is linearly-independent by A3,A5,A11;
hence thesis by A10;
end;
W is linearly-independent
proof let l be Linear_Combination of W;
assume that A12: Sum(l) = 0.V and A13: Carrier(l) <> {};
deffunc F(set)={C where C is Subset of V: $1 in C
& C in Z};
consider f being Function such that
A14: dom f = Carrier(l) and
A15: for x st x in Carrier(l) holds f.x =F(x) from Lambda;
reconsider M = rng f as non empty set by A13,A14,RELAT_1:65;
consider F being Choice_Function of M;
A16: now assume {} in M;
then consider x such that A17: x in dom f and A18: f.x = {}
by FUNCT_1:def 5;
Carrier(l) c= W by VECTSP_6:def 7;
then consider X such that
A19: x in X and A20: X in Z by A14,A17,TARSKI:def 4;
reconsider X as Subset of V by A3,A5,A20;
X in {C where C is Subset of V : x in C & C
in
Z}
by A19,A20;
hence contradiction by A14,A15,A17,A18;
end;
set S = rng F;
A21: dom F = M & M <> {} by A16,RLVECT_3:35;
then A22: S <> {} by RELAT_1:65;
A23: now let X;
assume X in S;
then consider x such that A24: x in dom F and A25: F.x = X
by FUNCT_1:def 5;
consider y such that A26: y in dom f and A27: f.y = x
by A21,A24,FUNCT_1:def 5;
A28: X in x by A16,A21,A24,A25,ORDERS_1:def 1;
x = {C where C is Subset of V :
y in C & C in Z} by A14,A15,A26,A27;
then ex C being Subset of V st
C = X & y in C & C in Z by A28;
hence X in Z;
end;
A29: now let X,Y;
assume X in S & Y in S;
then X in Z & Y in Z by A23;
then X,Y are_c=-comparable by A6,ORDINAL1:def 9;
hence X c= Y or Y c= X by XBOOLE_0:def 9;
end;
dom F is finite by A14,A21,FINSET_1:26;
then S is finite by FINSET_1:26;
then union S in S by A22,A29,RLVECT_3:34;
then union S in Z by A23;
then consider B being Subset of V such that
A30: B = union S and B c= A and
A31: B is linearly-independent by A3,A5;
Carrier(l) c= union S
proof let x;
assume A32: x in Carrier(l);
then A33: f.x in M by A14,FUNCT_1:def 5;
set X = f.x;
A34: F.X in X by A16,A33,ORDERS_1:def 1;
f.x = {C where C is Subset of V :
x in C & C in Z} by A15,A32;
then A35: ex C being Subset of V st
F.X = C & x in C & C in Z by A34;
F.X in S by A21,A33,FUNCT_1:def 5;
hence x in union S by A35,TARSKI:def 4;
end;
then l is Linear_Combination of B by A30,VECTSP_6:def 7;
hence thesis by A12,A13,A31,LMOD_5:def 1;
end;
hence union Z in Q by A3,A9;
end;
then consider X such that A36: X in Q and
A37: for Z st Z in Q & Z <> X holds not X c= Z by A4,ORDERS_2:79;
consider B being Subset of V such that
A38: B = X and A39: B c= A and
A40: B is linearly-independent by A3,A36;
take B;
thus B c= A & B is linearly-independent by A39,A40;
assume A41: Lin(B) <> the VectSpStr of V;
now assume A42: for v being Vector of V st v in A holds v in Lin(B);
now let v be Vector of V;
assume v in Lin(A);
then consider l being Linear_Combination of A such that
A43: v = Sum(l) by Th11;
A44: Carrier(l) c= the carrier of Lin(B)
proof let x;
assume A45: x in Carrier(l);
then reconsider a = x as Vector of V;
Carrier(l) c= A by VECTSP_6:def 7;
then a in Lin(B) by A42,A45;
hence thesis by RLVECT_1:def 1;
end;
reconsider F = the carrier of Lin(B) as
Subset of V by VECTSP_4:def 2;
reconsider l as Linear_Combination of F by A44,VECTSP_6:def 7;
Sum(l) = v by A43;
then v in Lin(F) by Th11;
hence v in Lin(B) by A1,Th15;
end;
then Lin(A) is Subspace of Lin(B) by VECTSP_4:36;
hence contradiction by A2,A41,VECTSP_4:33;
end;
then consider v being Vector of V such that
A46: v in A and A47: not v in Lin(B);
A48: B \/ {v} is linearly-independent
proof let l be Linear_Combination of B \/ {v};
assume A49: Sum(l) = 0.V;
now per cases;
suppose A50: v in Carrier(l);
deffunc F(Vector of V) = l.$1;
consider f being Function of the carrier of V,
the carrier of R such that A51: f.v = 0.R and
A52: for u being Vector of V st u <> v
holds f.u = F(u) from LambdaSep1;
reconsider f as Element of
Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
now let u be Vector of V;
assume not u in Carrier(l) \ {v};
then not u in Carrier(l) or u in {v} by XBOOLE_0:def 4;
then (l.u = 0.R & u <> v) or u = v by TARSKI:def 1,VECTSP_6:
20;
hence f.u = 0.R by A51,A52;
end;
then reconsider f as Linear_Combination of V by VECTSP_6:def 4;
Carrier(f) c= B
proof let x;
assume x in Carrier(f);
then x in {u where u is Vector of V: f.u <> 0.R}
by VECTSP_6:def 5;
then consider u being Vector of V such that
A53: u = x and A54: f.u <> 0.R;
f.u = l.u by A51,A52,A54;
then u in {v1 where v1 is Vector of V : l.v1 <> 0.R}
by A54;
then u in Carrier(l) & Carrier(l) c= B \/ {v}
by VECTSP_6:def 5,def 7;
then u in B or u in {v} by XBOOLE_0:def 2;
hence thesis by A51,A53,A54,TARSKI:def 1;
end;
then reconsider f as Linear_Combination of B by VECTSP_6:def 7;
deffunc F(Vector of V) = 0.R;
consider g being Function of the carrier of V,
the carrier of R such that A55: g.v = - l.v and
A56: for u being Vector of V st u <> v holds g.u = F(u)
from LambdaSep1;
reconsider g as Element of
Funcs(the carrier of V, the carrier of R) by FUNCT_2:11;
now let u be Vector of V;
assume not u in {v};
then u <> v by TARSKI:def 1;
hence g.u = 0.R by A56;
end;
then reconsider g as Linear_Combination of V by VECTSP_6:def 4;
Carrier(g) c= {v}
proof let x;
assume x in Carrier(g);
then x in {u where u is Vector of V : g.u <> 0.R}
by VECTSP_6:def 5;
then ex u being Vector of V st x = u & g.u <> 0.R;
then x = v by A56;
hence thesis by TARSKI:def 1;
end;
then reconsider g as Linear_Combination of {v} by VECTSP_6:def 7;
A57: f - g = l
proof let u be Vector of V;
now per cases;
suppose A58: v = u;
thus (f - g).u = f.u - g.u by VECTSP_6:73
.= 0.R + (- (- l.v))
by A51,A55,A58,RLVECT_1:def 11
.= l.v + 0.R by RLVECT_1:30
.= l.u by A58,RLVECT_1:10;
suppose A59: v <> u;
thus (f - g).u = f.u - g.u by VECTSP_6:73
.= l.u - g.u by A52,A59
.= l.u - 0.R by A56,A59
.= l.u by RLVECT_1:26;
end;
hence thesis;
end;
A60: Sum(g) = (- l.v) * v by A55,VECTSP_6:43;
0.V = Sum(f) - Sum(g) by A49,A57,VECTSP_6:80;
then Sum(f) = (- l.v) * v by A60,VECTSP_1:66;
then A61: (- l.v) * v in Lin(B) by Th11;
l.v <> 0.R by A50,VECTSP_6:20;
then - l.v <> 0.R by Lm1;
then (- l.v)" * ((- l.v) * v) = 1_ R * v by Lm2
.= v by VECTSP_1:def 26;
hence thesis by A47,A61,VECTSP_4:29;
suppose A62: not v in Carrier(l);
Carrier(l) c= B
proof let x;
assume A63: x in Carrier(l);
Carrier(l) c= B \/ {v} by VECTSP_6:def 7;
then x in B or x in {v} by A63,XBOOLE_0:def 2;
hence thesis by A62,A63,TARSKI:def 1;
end;
then l is Linear_Combination of B by VECTSP_6:def 7;
hence thesis by A40,A49,LMOD_5:def 1;
end;
hence thesis;
end;
v in {v} by TARSKI:def 1;
then A64: v in B \/ {v} & not v in B by A47,Th12,XBOOLE_0:def 2;
A65: B c= B \/ {v} by XBOOLE_1:7;
{v} c= A by A46,ZFMISC_1:37;
then B \/ {v} c= A by A39,XBOOLE_1:8;
then B \/ {v} in Q by A3,A48;
hence contradiction by A37,A38,A64,A65;
end;
Lm3: for V being LeftMod of R ex B being Subset of V st
B is base
proof let V be LeftMod of R;
{}(the carrier of V) is linearly-independent by LMOD_5:4;
then ex B being Subset of V st
{}(the carrier of V) c= B & B is base by Th26;
hence thesis;
end;
theorem
for V being LeftMod of R holds V is free
proof let V be LeftMod of R;
ex B being Subset of V st B is base by Lm3;
hence thesis by Def3;
end;
definition let R; let V be LeftMod of R;
canceled;
mode Basis of V -> Subset of V means
:Def5: it is base;
existence by Lm3;
end;
theorem
for V being LeftMod of R
for A being Subset of V holds
A is linearly-independent implies ex I being Basis of V st A c= I
proof let V be LeftMod of R;
let A be Subset of V;
assume A is linearly-independent;
then consider B being Subset of V such that A1: A c= B and
A2: B is base by Th26;
reconsider B as Basis of V by A2,Def5;
take B;
thus thesis by A1;
end;
theorem
for V being LeftMod of R
for A being Subset of V
holds Lin(A) = V implies ex I being Basis of V st I c= A
proof let V be LeftMod of R;
let A be Subset of V;
assume Lin(A) = V;
then consider B being Subset of V such that A1: B c= A and
A2: B is base by Th27;
reconsider B as Basis of V by A2,Def5;
take B;
thus thesis by A1;
end;