:: Basis of Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received July 10, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

Lm1: for V being RealLinearSpace
for F, G being FinSequence of the carrier of V
for f being Function of the carrier of V,REAL holds f (#) (F ^ G) = (f (#) F) ^ (f (#) G)

proof end;

theorem Th1: :: RLVECT_3:1

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds Sum (L1 + L2) = (Sum L1) + (Sum L2)

proof end;

theorem Th2: :: RLVECT_3:2

for a being Real
for V being RealLinearSpace
for L being Linear_Combination of V holds Sum (a * L) = a * (Sum L)

proof end;

reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

theorem Th3: :: RLVECT_3:3

for V being RealLinearSpace
for L being Linear_Combination of V holds Sum (- L) = - (Sum L)

proof end;

theorem Th4: :: RLVECT_3:4

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds Sum (L1 - L2) = (Sum L1) - (Sum L2)

proof end;

definition

let V be RealLinearSpace;
let A be Subset of V;

attr A is linearly-independent means :: RLVECT_3:def 1
for l being Linear_Combination of A st Sum l = 0. V holds
Carrier l = {} ;

end;

:: deftheorem defines linearly-independent RLVECT_3:def 1 :
for V being RealLinearSpace
for A being Subset of V holds
( A is linearly-independent iff for l being Linear_Combination of A st Sum l = 0. V holds
Carrier l = {} );

notation

let V be RealLinearSpace;
let A be Subset of V;
antonym linearly-dependent A for linearly-independent ;

end;

theorem :: RLVECT_3:5

for V being RealLinearSpace
for A, B being Subset of V st A c= B & B is linearly-independent holds
A is linearly-independent

proof end;

theorem Th6: :: RLVECT_3:6

for V being RealLinearSpace
for A being Subset of V st A is linearly-independent holds
not 0. V in A

proof end;

theorem Th7: :: RLVECT_3:7

for V being RealLinearSpace holds {} the carrier of V is linearly-independent

proof end;

registration

let V be RealLinearSpace;

cluster linearly-independent for Element of bool the carrier of V;

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

proof end;

end;

theorem Th8: :: RLVECT_3:8

for V being RealLinearSpace
for v being VECTOR of V holds
( {v} is linearly-independent iff v <> 0. V )

proof end;

theorem :: RLVECT_3:9

for V being RealLinearSpace holds {(0. V)} is linearly-dependent by Th8;

theorem Th10: :: RLVECT_3:10

for V being RealLinearSpace
for v1, v2 being VECTOR of V st {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )

proof end;

theorem :: RLVECT_3:11

for V being RealLinearSpace
for v being VECTOR of V holds
( {v,(0. V)} is linearly-dependent & {(0. V),v} is linearly-dependent ) by Th10;

theorem Th12: :: RLVECT_3:12

for V being RealLinearSpace
for v1, v2 being VECTOR of V holds
( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a being Real holds v1 <> a * v2 ) ) )

proof end;

theorem :: RLVECT_3:13

for V being RealLinearSpace
for v1, v2 being VECTOR of V holds
( ( v1 <> v2 & {v1,v2} is linearly-independent ) iff for a, b being Real st (a * v1) + (b * v2) = 0. V holds
( a = 0 & b = 0 ) )

proof end;

definition

let V be RealLinearSpace;
let A be Subset of V;

func Lin A -> strict Subspace of V means :Def2: :: RLVECT_3:def 2
the carrier of it = { (Sum l) where l is Linear_Combination of A : verum } ;

existence
ex b₁ being strict Subspace of V st the carrier of b₁ = { (Sum l) where l is Linear_Combination of A : verum }

proof end;

uniqueness
for b₁, b₂ being strict Subspace of V st the carrier of b₁ = { (Sum l) where l is Linear_Combination of A : verum } & the carrier of b₂ = { (Sum l) where l is Linear_Combination of A : verum } holds
b₁ = b₂ by RLSUB_1:30;

end;

:: deftheorem Def2 defines Lin RLVECT_3:def 2 :
for V being RealLinearSpace
for A being Subset of V
for b₃ being strict Subspace of V holds
( b₃ = Lin A iff the carrier of b₃ = { (Sum l) where l is Linear_Combination of A : verum } );

theorem Th14: :: RLVECT_3:14

for x being object
for V being RealLinearSpace
for A being Subset of V holds
( x in Lin A iff ex l being Linear_Combination of A st x = Sum l )

proof end;

theorem Th15: :: RLVECT_3:15

for x being object
for V being RealLinearSpace
for A being Subset of V st x in A holds
x in Lin A

proof end;

Lm2: for x being object
for V being RealLinearSpace holds
( x in (0). V iff x = 0. V )

proof end;

theorem :: RLVECT_3:16

for V being RealLinearSpace holds Lin ({} the carrier of V) = (0). V

proof end;

theorem :: RLVECT_3:17

for V being RealLinearSpace
for A being Subset of V holds
( not Lin A = (0). V or A = {} or A = {(0. V)} )

proof end;

theorem Th18: :: RLVECT_3:18

for V being RealLinearSpace
for A being Subset of V
for W being strict Subspace of V st A = the carrier of W holds
Lin A = W

proof end;

theorem :: RLVECT_3:19

for V being strict RealLinearSpace
for A being Subset of V st A = the carrier of V holds
Lin A = V

proof end;

Lm3: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W3 holds
W1 /\ W2 is Subspace of W3

proof end;

Lm4: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 & W1 is Subspace of W3 holds
W1 is Subspace of W2 /\ W3

proof end;

Lm5: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W1 is Subspace of W2 + W3

proof end;

Lm6: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W3 & W2 is Subspace of W3 holds
W1 + W2 is Subspace of W3

proof end;

theorem Th20: :: RLVECT_3:20

for V being RealLinearSpace
for A, B being Subset of V st A c= B holds
Lin A is Subspace of Lin B

proof end;

theorem :: RLVECT_3:21

for V being strict RealLinearSpace
for A, B being Subset of V st Lin A = V & A c= B holds
Lin B = V

proof end;

theorem :: RLVECT_3:22

for V being RealLinearSpace
for A, B being Subset of V holds Lin (A \/ B) = (Lin A) + (Lin B)

proof end;

theorem :: RLVECT_3:23

for V being RealLinearSpace
for A, B being Subset of V holds Lin (A /\ B) is Subspace of (Lin A) /\ (Lin B)

proof end;

Lm7: for M being non empty set
for CF being Choice_Function of M st not {} in M holds
dom CF = M

proof end;

theorem Th24: :: RLVECT_3:24

for V being RealLinearSpace
for A being Subset of V st A is linearly-independent holds
ex B being Subset of V st
( A c= B & B is linearly-independent & Lin B = RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) )

proof end;

theorem Th25: :: RLVECT_3:25

for V being RealLinearSpace
for A being Subset of V st Lin A = V holds
ex B being Subset of V st
( B c= A & B is linearly-independent & Lin B = V )

proof end;

definition

let V be RealLinearSpace;

mode Basis of V -> Subset of V means :Def3: :: RLVECT_3:def 3
( it is linearly-independent & Lin it = RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) );

existence
ex b₁ being Subset of V st
( b₁ is linearly-independent & Lin b₁ = RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) )

proof end;

end;

:: deftheorem Def3 defines Basis RLVECT_3:def 3 :
for V being RealLinearSpace
for b₂ being Subset of V holds
( b₂ is Basis of V iff ( b₂ is linearly-independent & Lin b₂ = RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) ) );

theorem :: RLVECT_3:26

for V being strict 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 :: RLVECT_3:27

for V being RealLinearSpace
for A being Subset of V st Lin A = V holds
ex I being Basis of V st I c= A

proof end;

::
:: Auxiliary theorems.
::

theorem :: RLVECT_3:28

for M being non empty set
for CF being Choice_Function of M st not {} in M holds
dom CF = M by Lm7;

theorem :: RLVECT_3:29

for x being object
for V being RealLinearSpace holds
( x in (0). V iff x = 0. V ) by Lm2;

theorem :: RLVECT_3:30

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W3 holds
W1 /\ W2 is Subspace of W3 by Lm3;

theorem :: RLVECT_3:31

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 & W1 is Subspace of W3 holds
W1 is Subspace of W2 /\ W3 by Lm4;

theorem :: RLVECT_3:32

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W3 & W2 is Subspace of W3 holds
W1 + W2 is Subspace of W3 by Lm6;

theorem :: RLVECT_3:33

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W1 is Subspace of W2 + W3 by Lm5;

theorem :: RLVECT_3:34

for V being RealLinearSpace
for F, G being FinSequence of the carrier of V
for f being Function of the carrier of V,REAL holds f (#) (F ^ G) = (f (#) F) ^ (f (#) G) by Lm1;