:: The Steinitz Theorem and the Dimension of a Real Linear Space
:: by JingChao Chen
::
:: Received July 1, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users

theorem Th1: :: RLVECT_5:1

for V being RealLinearSpace
for KL1, KL2 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds
KL1 = KL2

proof end;

theorem Th2: :: RLVECT_5:2

for V being RealLinearSpace
for A being Subset of V st A is linearly-independent holds
ex I being Basis of V st A c= I

proof end;

theorem Th3: :: RLVECT_5:3

for V being RealLinearSpace
for L being Linear_Combination of V
for x being VECTOR of V holds
( x in Carrier L iff ex v being VECTOR of V st
( x = v & L . v <> 0 ) )

proof end;

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

proof end;

:: More On Linear Combinations

theorem Th4: :: RLVECT_5:4

for V being RealLinearSpace
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 Th5: :: RLVECT_5:5

for V being RealLinearSpace
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 Th6: :: RLVECT_5:6

for V being RealLinearSpace
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 Th7: :: RLVECT_5:7

for V being RealLinearSpace
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 Th8: :: RLVECT_5:8

for V being RealLinearSpace
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 Th9: :: RLVECT_5:9

for V being RealLinearSpace
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 Th10: :: RLVECT_5:10

for V being RealLinearSpace
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 Th11: :: RLVECT_5:11

for V being RealLinearSpace
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 Th12: :: RLVECT_5:12

for V being RealLinearSpace
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 Th13: :: RLVECT_5:13

for V being RealLinearSpace
for I being Basis of V
for v being VECTOR of V holds v in Lin I

proof end;

theorem Th14: :: RLVECT_5:14

for V being RealLinearSpace
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 Th15: :: RLVECT_5:15

for V being RealLinearSpace
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 Th16: :: RLVECT_5:16

for V being RealLinearSpace
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 Th17: :: RLVECT_5:17

for V being RealLinearSpace
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 Th18: :: RLVECT_5:18

for V being RealLinearSpace
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 Th19: :: RLVECT_5:19

for V being RealLinearSpace
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 Th20: :: RLVECT_5:20

for V being RealLinearSpace
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;

:: Exchange Lemma

theorem Th21: :: RLVECT_5:21

for V being RealLinearSpace
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 Th22: :: RLVECT_5:22

for V being RealLinearSpace
for A, B being finite Subset of V st RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult 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) & RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) = Lin (B \/ C) ) )

proof end;

definition

let V be RealLinearSpace;

attr V is finite-dimensional means :Def1: :: RLVECT_5:def 1
ex A being finite Subset of V st A is Basis of V;

end;

:: deftheorem Def1 defines finite-dimensional RLVECT_5:def 1 :
for V being RealLinearSpace holds
( V is finite-dimensional iff ex A being finite Subset of V st A is Basis of V );

registration

cluster non empty V100() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional for RLSStruct ;

existence
ex b₁ being RealLinearSpace st
( b₁ is strict & b₁ is finite-dimensional )

proof end;

end;

theorem Th23: :: RLVECT_5:23

for V being RealLinearSpace st V is finite-dimensional holds
for I being Basis of V holds I is finite

proof end;

theorem :: RLVECT_5:24

for V being RealLinearSpace st V is finite-dimensional holds
for A being Subset of V st A is linearly-independent holds
A is finite

proof end;

theorem Th25: :: RLVECT_5:25

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

proof end;

theorem Th26: :: RLVECT_5:26

for V being RealLinearSpace holds (0). V is finite-dimensional

proof end;

theorem Th27: :: RLVECT_5:27

for V being RealLinearSpace
for W being Subspace of V st V is finite-dimensional holds
W is finite-dimensional

proof end;

registration

let V be RealLinearSpace;

cluster non empty V100() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional for Subspace of V;

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

proof end;

end;

registration

let V be finite-dimensional RealLinearSpace;

cluster -> finite-dimensional for Subspace of V;

correctness
coherence
for b₁ being Subspace of V holds b₁ is finite-dimensional ;
by Th27;

end;

registration

let V be finite-dimensional RealLinearSpace;

cluster non empty V100() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional for Subspace of V;

existence
ex b₁ being Subspace of V st b₁ is strict

proof end;

end;

definition

let V be RealLinearSpace;
assume A1: V is finite-dimensional ;

func dim V -> Element of NAT means :Def2: :: RLVECT_5:def 2
for I being Basis of V holds it = card I;

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

proof end;

uniqueness
for b₁, b₂ being Element of 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 Def2 defines dim RLVECT_5:def 2 :
for V being RealLinearSpace st V is finite-dimensional holds
for b₂ being Element of NAT holds
( b₂ = dim V iff for I being Basis of V holds b₂ = card I );

theorem Th28: :: RLVECT_5:28

for V being finite-dimensional RealLinearSpace
for W being Subspace of V holds dim W <= dim V

proof end;

theorem Th29: :: RLVECT_5:29

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

proof end;

theorem Th30: :: RLVECT_5:30

for V being finite-dimensional RealLinearSpace holds dim V = dim ((Omega). V)

proof end;

theorem :: RLVECT_5:31

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

proof end;

theorem Th32: :: RLVECT_5:32

for V being finite-dimensional RealLinearSpace holds
( dim V = 0 iff (Omega). V = (0). V )

proof end;

theorem :: RLVECT_5:33

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

proof end;

theorem :: RLVECT_5:34

for V being finite-dimensional RealLinearSpace 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 Th35: :: RLVECT_5:35

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

proof end;

theorem :: RLVECT_5:36

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

proof end;

theorem :: RLVECT_5:37

for V being finite-dimensional RealLinearSpace
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 n being Element of NAT
for V being finite-dimensional RealLinearSpace st n <= dim V holds
ex W being strict Subspace of V st dim W = n

proof end;

theorem :: RLVECT_5:38

for n being Element of NAT
for V being finite-dimensional RealLinearSpace holds
( n <= dim V iff ex W being strict Subspace of V st dim W = n ) by Lm2, Th28;

definition

let V be finite-dimensional RealLinearSpace;
let n be Element of NAT ;

func n Subspaces_of V -> set means :Def3: :: RLVECT_5:def 3
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 Def3 defines Subspaces_of RLVECT_5:def 3 :
for V being finite-dimensional RealLinearSpace
for n being Element of NAT
for b₃ being set holds
( b₃ = n Subspaces_of V iff for x being object holds
( x in b₃ iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) );

theorem :: RLVECT_5:39

for n being Element of NAT
for V being finite-dimensional RealLinearSpace st n <= dim V holds
not n Subspaces_of V is empty

proof end;

theorem :: RLVECT_5:40

for n being Element of NAT
for V being finite-dimensional RealLinearSpace st dim V < n holds
n Subspaces_of V = {}

proof end;

theorem :: RLVECT_5:41

for n being Element of NAT
for V being finite-dimensional RealLinearSpace
for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V

proof end;