:: Steinitz Theorem and Dimension of a Vector Space
:: by Mariusz \.Zynel
::
:: Received October 6, 1995
:: Copyright (c) 1995-2021 Association of Mizar Users

::
:: Preliminaries
::

registration

let S be non empty 1-sorted ;

cluster non empty for Element of K1( the carrier of S);

existence
not for b₁ being Subset of S holds b₁ is empty

proof end;

end;

Lm1: for X, x being set st x in X holds
(X \ {x}) \/ {x} = X

proof end;

:: More On Linear Combinations

theorem Th1: :: VECTSP_9:1

for GF being Field
for V being VectSp of GF
for L being Linear_Combination of V
for F, G being FinSequence of the carrier of V
for P being Permutation of (dom F) st G = F * P holds
Sum (L (#) F) = Sum (L (#) G)

proof end;

theorem Th2: :: VECTSP_9:2

for GF being Field
for V being VectSp of GF
for L being Linear_Combination of V
for F being FinSequence of the carrier of V st Carrier L misses rng F holds
Sum (L (#) F) = 0. V

proof end;

theorem Th3: :: VECTSP_9:3

for GF being Field
for V being VectSp of GF
for F being FinSequence of the carrier of V st F is one-to-one holds
for L being Linear_Combination of V st Carrier L c= rng F holds
Sum (L (#) F) = Sum L

proof end;

theorem Th4: :: VECTSP_9:4

for GF being Field
for V being VectSp of GF
for L being Linear_Combination of V
for F being FinSequence of the carrier of V ex K being Linear_Combination of V st
( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F )

proof end;

theorem Th5: :: VECTSP_9:5

for GF being Field
for V being VectSp of GF
for L being Linear_Combination of V
for A being Subset of V
for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds
ex K being Linear_Combination of A st Sum (L (#) F) = Sum K

proof end;

theorem Th6: :: VECTSP_9:6

for GF being Field
for V being VectSp of GF
for L being Linear_Combination of V
for A being Subset of V st Carrier L c= the carrier of (Lin A) holds
ex K being Linear_Combination of A st Sum L = Sum K

proof end;

theorem Th7: :: VECTSP_9:7

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for L being Linear_Combination of V st Carrier L c= the carrier of W holds
for K being Linear_Combination of W st K = L | the carrier of W holds
( Carrier L = Carrier K & Sum L = Sum K )

proof end;

theorem Th8: :: VECTSP_9:8

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for K being Linear_Combination of W ex L being Linear_Combination of V st
( Carrier K = Carrier L & Sum K = Sum L )

proof end;

theorem Th9: :: VECTSP_9:9

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for L being Linear_Combination of V st Carrier L c= the carrier of W holds
ex K being Linear_Combination of W st
( Carrier K = Carrier L & Sum K = Sum L )

proof end;

:: More On Linear Independence & Basis

theorem Th10: :: VECTSP_9:10

for GF being Field
for V being VectSp of GF
for I being Basis of V
for v being Vector of V holds v in Lin I

proof end;

registration

let GF be Field;
let V be VectSp of GF;

cluster linearly-independent for Element of K1( the carrier of V);

existence
ex b₁ being Subset of V st b₁ is linearly-independent

proof end;

end;

theorem Th11: :: VECTSP_9:11

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for A being Subset of W st A is linearly-independent holds
A is linearly-independent Subset of V

proof end;

theorem Th12: :: VECTSP_9:12

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for A being Subset of V st A is linearly-independent & A c= the carrier of W holds
A is linearly-independent Subset of W

proof end;

theorem Th13: :: VECTSP_9:13

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for A being Basis of W ex B being Basis of V st A c= B

proof end;

theorem Th14: :: VECTSP_9:14

for GF being Field
for V being VectSp of GF
for A being Subset of V st A is linearly-independent holds
for v being Vector of V st v in A holds
for B being Subset of V st B = A \ {v} holds
not v in Lin B

proof end;

theorem Th15: :: VECTSP_9:15

for GF being Field
for V being VectSp of GF
for I being Basis of V
for A being non empty Subset of V st A misses I holds
for B being Subset of V st B = I \/ A holds
B is linearly-dependent

proof end;

theorem Th16: :: VECTSP_9:16

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for A being Subset of V st A c= the carrier of W holds
Lin A is Subspace of W

proof end;

theorem Th17: :: VECTSP_9:17

for GF being Field
for V being VectSp of GF
for W being Subspace of V
for A being Subset of V
for B being Subset of W st A = B holds
Lin A = Lin B

proof end;

::
:: Steinitz Theorem
::
:: Exchange Lemma

theorem Th18: :: VECTSP_9:18

for GF being Field
for V being VectSp of GF
for A, B being finite Subset of V
for v being Vector of V st v in Lin (A \/ B) & not v in Lin B holds
ex w being Vector of V st
( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) )

proof end;

:: Steinitz Theorem

theorem Th19: :: VECTSP_9:19

for GF being Field
for V being VectSp of GF
for A, B being finite Subset of V st ModuleStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent holds
( card B <= card A & ex C being finite Subset of V st
( C c= A & card C = (card A) - (card B) & ModuleStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) )

proof end;

::
:: Finite-Dimensional Vector Spaces
::

theorem Th20: :: VECTSP_9:20

for GF being Field
for V being VectSp of GF st V is finite-dimensional holds
for I being Basis of V holds I is finite

proof end;

theorem :: VECTSP_9:21

for GF being Field
for V being VectSp of GF st V is finite-dimensional holds
for A being Subset of V st A is linearly-independent holds
A is finite

proof end;

theorem Th22: :: VECTSP_9:22

for GF being Field
for V being VectSp of GF st V is finite-dimensional holds
for A, B being Basis of V holds card A = card B

proof end;

theorem Th23: :: VECTSP_9:23

for GF being Field
for V being VectSp of GF holds (0). V is finite-dimensional

proof end;

theorem Th24: :: VECTSP_9:24

for GF being Field
for V being VectSp of GF
for W being Subspace of V st V is finite-dimensional holds
W is finite-dimensional

proof end;

registration

let GF be Field;
let V be VectSp of GF;

cluster non empty V88() strict V119(GF) V120(GF) V121(GF) V122(GF) V127() V128() V129() finite-dimensional for Subspace of V;

existence
ex b₁ being Subspace of V st
( b₁ is strict & b₁ is finite-dimensional )

proof end;

end;

registration

let GF be Field;
let V be finite-dimensional VectSp of GF;

cluster -> finite-dimensional for Subspace of V;

correctness
coherence
for b₁ being Subspace of V holds b₁ is finite-dimensional ;
by Th24;

end;

registration

let GF be Field;
let V be finite-dimensional VectSp of GF;

cluster non empty V88() strict V119(GF) V120(GF) V121(GF) V122(GF) V127() V128() V129() finite-dimensional for Subspace of V;

existence
ex b₁ being Subspace of V st b₁ is strict

proof end;

end;

::
:: Dimension of a Vector Space
::

definition

let GF be Field;
let V be VectSp of GF;
assume A1: V is finite-dimensional ;

func dim V -> Nat means :Def1: :: VECTSP_9:def 1
for I being Basis of V holds it = card I;

existence
ex b₁ being Nat st
for I being Basis of V holds b₁ = card I

proof end;

uniqueness
for b₁, b₂ being Nat st ( for I being Basis of V holds b₁ = card I ) & ( for I being Basis of V holds b₂ = card I ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines dim VECTSP_9:def 1 :
for GF being Field
for V being VectSp of GF st V is finite-dimensional holds
for b₃ being Nat holds
( b₃ = dim V iff for I being Basis of V holds b₃ = card I );

theorem Th25: :: VECTSP_9:25

for GF being Field
for V being finite-dimensional VectSp of GF
for W being Subspace of V holds dim W <= dim V

proof end;

theorem Th26: :: VECTSP_9:26

for GF being Field
for V being finite-dimensional VectSp of GF
for A being Subset of V st A is linearly-independent holds
card A = dim (Lin A)

proof end;

theorem Th27: :: VECTSP_9:27

for GF being Field
for V being finite-dimensional VectSp of GF holds dim V = dim ((Omega). V)

proof end;

theorem :: VECTSP_9:28

for GF being Field
for V being finite-dimensional VectSp of GF
for W being Subspace of V holds
( dim V = dim W iff (Omega). V = (Omega). W )

proof end;

theorem Th29: :: VECTSP_9:29

for GF being Field
for V being finite-dimensional VectSp of GF holds
( dim V = 0 iff (Omega). V = (0). V )

proof end;

theorem :: VECTSP_9:30

for GF being Field
for V being finite-dimensional VectSp of GF holds
( dim V = 1 iff ex v being Vector of V st
( v <> 0. V & (Omega). V = Lin {v} ) )

proof end;

theorem :: VECTSP_9:31

for GF being Field
for V being finite-dimensional VectSp of GF holds
( dim V = 2 iff ex u, v being Vector of V st
( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) )

proof end;

theorem Th32: :: VECTSP_9:32

for GF being Field
for V being finite-dimensional VectSp of GF
for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2)

proof end;

theorem :: VECTSP_9:33

for GF being Field
for V being finite-dimensional VectSp of GF
for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V)

proof end;

theorem :: VECTSP_9:34

for GF being Field
for V being finite-dimensional VectSp of GF
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
dim V = (dim W1) + (dim W2)

proof end;

Lm2: for GF being Field
for n being Nat
for V being finite-dimensional VectSp of GF st n <= dim V holds
ex W being strict Subspace of V st dim W = n

proof end;

theorem :: VECTSP_9:35

for GF being Field
for n being Nat
for V being finite-dimensional VectSp of GF holds
( n <= dim V iff ex W being strict Subspace of V st dim W = n ) by Lm2, Th25;

definition

let GF be Field;
let V be finite-dimensional VectSp of GF;
let n be Nat;

func n Subspaces_of V -> set means :Def2: :: VECTSP_9:def 2
for x being object holds
( x in it iff ex W being strict Subspace of V st
( W = x & dim W = n ) );

existence
ex b₁ being set st
for x being object holds
( x in b₁ iff ex W being strict Subspace of V st
( W = x & dim W = n ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being object holds
( x in b₁ iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) ) & ( for x being object holds
( x in b₂ iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines Subspaces_of VECTSP_9:def 2 :
for GF being Field
for V being finite-dimensional VectSp of GF
for n being Nat
for b₄ being set holds
( b₄ = n Subspaces_of V iff for x being object holds
( x in b₄ iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) );

theorem :: VECTSP_9:36

for GF being Field
for n being Nat
for V being finite-dimensional VectSp of GF st n <= dim V holds
not n Subspaces_of V is empty

proof end;

theorem :: VECTSP_9:37

for GF being Field
for n being Nat
for V being finite-dimensional VectSp of GF st dim V < n holds
n Subspaces_of V = {}

proof end;

theorem :: VECTSP_9:38

for GF being Field
for n being Nat
for V being finite-dimensional VectSp of GF
for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V

proof end;