:: VECTSP_5 semantic presentation

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let M be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
let W1, W2 be Subspace of M;
func W1 + W2 -> strict Subspace of M means :Def1: :: VECTSP_5:def 1
the carrier of it = { (v + u) where v, u is Element of M : ( v in W1 & u in W2 ) } ;
existence

proof end;

uniqueness by VECTSP_4:37;

end;

:: deftheorem Def1 defines + VECTSP_5:def 1 :

Lm1: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds W1 + W2 = W2 + W1

proof end;

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let M be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
let W1, W2 be Subspace of M;
func W1 /\ W2 -> strict Subspace of M means :Def2: :: VECTSP_5:def 2
the carrier of it = the carrier of W1 /\ the carrier of W2;
existence

proof end;

uniqueness by VECTSP_4:37;
commutativity ;

end;

:: deftheorem Def2 defines /\ VECTSP_5:def 2 :

theorem :: VECTSP_5:1

canceled;

theorem :: VECTSP_5:2

canceled;

theorem :: VECTSP_5:3

canceled;

theorem :: VECTSP_5:4

canceled;

theorem Th5: :: VECTSP_5:5

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M
for x being set holds
( x in W1 + W2 iff ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

proof end;

theorem Th6: :: VECTSP_5:6

proof end;

theorem Th7: :: VECTSP_5:7

proof end;

Lm2: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of W1 c= the carrier of (W1 + W2)

proof end;

Lm3: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1 being Subspace of M
for W2 being strict Subspace of M st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2

proof end;

theorem :: VECTSP_5:8

theorem :: VECTSP_5:9

theorem Th10: :: VECTSP_5:10

proof end;

theorem Th11: :: VECTSP_5:11

proof end;

theorem Th12: :: VECTSP_5:12

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1 being Subspace of M
for W2 being strict Subspace of M holds
( W1 is Subspace of W2 iff W1 + W2 = W2 )

proof end;

theorem Th13: :: VECTSP_5:13

proof end;

Lm4: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W being Subspace of M ex W' being strict Subspace of M st the carrier of W = the carrier of W'

proof end;

Lm5: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W, W', W1 being Subspace of M st the carrier of W = the carrier of W' holds
( W1 + W = W1 + W' & W + W1 = W' + W1 )

proof end;

Lm6: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W being Subspace of M holds W is Subspace of (Omega). M

proof end;

theorem Th14: :: VECTSP_5:14

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF holds
( ((0). M) + ((Omega). M) = VectSpStr(# the carrier of M,the add of M,the Zero of M,the lmult of M #) & ((Omega). M) + ((0). M) = VectSpStr(# the carrier of M,the add of M,the Zero of M,the lmult of M #) ) by Th13;

theorem Th15: :: VECTSP_5:15

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W being Subspace of M holds
( ((Omega). M) + W = VectSpStr(# the carrier of M,the add of M,the Zero of M,the lmult of M #) & W + ((Omega). M) = VectSpStr(# the carrier of M,the add of M,the Zero of M,the lmult of M #) )

proof end;

theorem :: VECTSP_5:16

theorem :: VECTSP_5:17

proof end;

theorem :: VECTSP_5:18

canceled;

theorem Th19: :: VECTSP_5:19

proof end;

Lm7: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ W2) c= the carrier of W1

proof end;

theorem Th20: :: VECTSP_5:20

proof end;

Lm8: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W, W', W1 being Subspace of M st the carrier of W = the carrier of W' holds
( W1 /\ W = W1 /\ W' & W /\ W1 = W' /\ W1 )

proof end;

theorem Th21: :: VECTSP_5:21

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W2 being Subspace of M holds
( ( for W1 being strict Subspace of M st W1 is Subspace of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Subspace of M st W1 /\ W2 = W1 holds
W1 is Subspace of W2 ) )

proof end;

theorem :: VECTSP_5:22

proof end;

theorem :: VECTSP_5:23

proof end;

theorem :: VECTSP_5:24

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 & W1 is Subspace of W3 holds
W1 is Subspace of W2 /\ W3

proof end;

theorem Th25: :: VECTSP_5:25

proof end;

theorem :: VECTSP_5:26

canceled;

theorem Th27: :: VECTSP_5:27

proof end;

theorem :: VECTSP_5:28

Lm9: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)

proof end;

theorem :: VECTSP_5:29

proof end;

Lm10: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2

proof end;

theorem Th30: :: VECTSP_5:30

proof end;

Lm11: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1

proof end;

theorem Th31: :: VECTSP_5:31

proof end;

Lm12: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2, W3 being Subspace of M holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))

proof end;

theorem :: VECTSP_5:32

proof end;

Lm13: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))

proof end;

theorem Th33: :: VECTSP_5:33

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3)

proof end;

Lm14: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W2, W1, W3 being Subspace of M holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: VECTSP_5:34

proof end;

Lm15: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: VECTSP_5:35

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3)

proof end;

theorem Th36: :: VECTSP_5:36

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W3, W2 being Subspace of M
for W1 being strict Subspace of M st W1 is Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3

proof end;

theorem :: VECTSP_5:37

proof end;

theorem :: VECTSP_5:38

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1 being Subspace of M
for W2, W3 being strict Subspace of M st W1 is Subspace of W2 holds
W1 + W3 is Subspace of W2 + W3

proof end;

theorem :: VECTSP_5:39

proof end;

theorem :: VECTSP_5:40

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W3, W2 being Subspace of M st W1 is Subspace of W3 & W2 is Subspace of W3 holds
W1 + W2 is Subspace of W3

proof end;

theorem :: VECTSP_5:41

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds
( ( W1 is Subspace of W2 or W2 is Subspace of W1 ) iff ex W being Subspace of M st the carrier of W = the carrier of W1 \/ the carrier of W2 )

proof end;

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let M be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
func Subspaces M -> set means :Def3: :: VECTSP_5:def 3
for x being set holds
( x in it iff ex W being strict Subspace of M st W = x );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines Subspaces VECTSP_5:def 3 :

registration

proof end;

end;

theorem :: VECTSP_5:42

canceled;

theorem :: VECTSP_5:43

canceled;

theorem :: VECTSP_5:44

proof end;

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let M be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
let W1, W2 be Subspace of M;
pred M is_the_direct_sum_of W1,W2 means :Def4: :: VECTSP_5:def 4
( VectSpStr(# the carrier of M,the add of M,the Zero of M,the lmult of M #) = W1 + W2 & W1 /\ W2 = (0). M );

end;

:: deftheorem Def4 defines is_the_direct_sum_of VECTSP_5:def 4 :

Lm16: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds
( W1 + W2 = VectSpStr(# the carrier of M,the add of M,the Zero of M,the lmult of M #) iff for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )

proof end;

definition

let F be Field;
let V be VectSp of F;
let W be Subspace of V;
mode Linear_Compl of W -> Subspace of V means :Def5: :: VECTSP_5:def 5
V is_the_direct_sum_of it,W;
existence

proof end;

end;

:: deftheorem Def5 defines Linear_Compl VECTSP_5:def 5 :

Lm17: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M st M is_the_direct_sum_of W1,W2 holds
M is_the_direct_sum_of W2,W1

proof end;

theorem :: VECTSP_5:45

canceled;

theorem :: VECTSP_5:46

canceled;

theorem :: VECTSP_5:47

for F being Field
for V being VectSp of F
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1

proof end;

theorem Th48: :: VECTSP_5:48

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )

proof end;

theorem Th49: :: VECTSP_5:49

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = VectSpStr(# the carrier of V,the add of V,the Zero of V,the lmult of V #) & L + W = VectSpStr(# the carrier of V,the add of V,the Zero of V,the lmult of V #) )

proof end;

theorem Th50: :: VECTSP_5:50

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W holds
( W /\ L = (0). V & L /\ W = (0). V )

proof end;

theorem :: VECTSP_5:51

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M st M is_the_direct_sum_of W1,W2 holds
M is_the_direct_sum_of W2,W1 by Lm17;

theorem Th52: :: VECTSP_5:52

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF holds
( M is_the_direct_sum_of (0). M, (Omega). M & M is_the_direct_sum_of (Omega). M, (0). M )

proof end;

theorem :: VECTSP_5:53

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W holds W is Linear_Compl of L

proof end;

theorem :: VECTSP_5:54

for F being Field
for V being VectSp of F holds
( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V )

proof end;

theorem Th55: :: VECTSP_5:55

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2

proof end;

theorem Th56: :: VECTSP_5:56

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds
( M is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being Element of M st C1 /\ C2 = {v} )

proof end;

theorem :: VECTSP_5:57

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable strict VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M holds
( W1 + W2 = M iff for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) by Lm16;

theorem Th58: :: VECTSP_5:58

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M
for v, v1, v2, u1, u2 being Element of M st M is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )

proof end;

theorem :: VECTSP_5:59

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M st M = W1 + W2 & ex v being Element of M st
for v1, v2, u1, u2 being Element of M st v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
M is_the_direct_sum_of W1,W2

proof end;

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let M be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
let v be Element of M;
let W1, W2 be Subspace of M;
assume A1: M is_the_direct_sum_of W1,W2 ;
func v |-- W1,W2 -> Element of [:the carrier of M,the carrier of M:] means :Def6: :: VECTSP_5:def 6
( v = (it `1 ) + (it `2 ) & it `1 in W1 & it `2 in W2 );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines |-- VECTSP_5:def 6 :

theorem :: VECTSP_5:60

canceled;

theorem :: VECTSP_5:61

canceled;

theorem :: VECTSP_5:62

canceled;

theorem :: VECTSP_5:63

canceled;

theorem Th64: :: VECTSP_5:64

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M
for v being Element of M st M is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `1 = (v |-- W2,W1) `2

proof end;

theorem Th65: :: VECTSP_5:65

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for M being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for W1, W2 being Subspace of M
for v being Element of M st M is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1

proof end;

theorem :: VECTSP_5:66

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V
for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L

proof end;

theorem :: VECTSP_5:67

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V holds ((v |-- W,L) `1 ) + ((v |-- W,L) `2 ) = v

proof end;

theorem :: VECTSP_5:68

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V holds
( (v |-- W,L) `1 in W & (v |-- W,L) `2 in L )

proof end;

theorem :: VECTSP_5:69

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V holds (v |-- W,L) `1 = (v |-- L,W) `2

proof end;

theorem :: VECTSP_5:70

for F being Field
for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V holds (v |-- W,L) `2 = (v |-- L,W) `1

proof end;

definition