:: Operations on Subspaces in Vector Space
:: by Wojciech A. Trybulec
::
:: Copyright (c) 1990-2018 Association of Mizar Users

definition
let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 u, v is Element of M : ( v in W1 & u in W2 ) } ;
existence
ex b1 being strict Subspace of M st the carrier of b1 = { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) }
proof end;
uniqueness
for b1, b2 being strict Subspace of M st the carrier of b1 = { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } & the carrier of b2 = { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } holds
b1 = b2
by VECTSP_4:29;
end;

:: deftheorem Def1 defines + VECTSP_5:def 1 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M
for b5 being strict Subspace of M holds
( b5 = W1 + W2 iff the carrier of b5 = { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } );

Lm1: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds W1 + W2 = W2 + W1

proof end;

definition
let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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
ex b1 being strict Subspace of M st the carrier of b1 = the carrier of W1 /\ the carrier of W2
proof end;
uniqueness
for b1, b2 being strict Subspace of M st the carrier of b1 = the carrier of W1 /\ the carrier of W2 & the carrier of b2 = the carrier of W1 /\ the carrier of W2 holds
b1 = b2
by VECTSP_4:29;
commutativity
for b1 being strict Subspace of M
for W1, W2 being Subspace of M st the carrier of b1 = the carrier of W1 /\ the carrier of W2 holds
the carrier of b1 = the carrier of W2 /\ the carrier of W1
;
end;

:: deftheorem Def2 defines /\ VECTSP_5:def 2 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M
for b5 being strict Subspace of M holds
( b5 = W1 /\ W2 iff the carrier of b5 = the carrier of W1 /\ the carrier of W2 );

theorem Th1: :: VECTSP_5:1
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M
for x being object 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 Th2: :: VECTSP_5:2
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M
for v being Element of M st ( v in W1 or v in W2 ) holds
v in W1 + W2
proof end;

theorem Th3: :: VECTSP_5:3
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M
for x being object holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
proof end;

Lm2: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:4

theorem :: VECTSP_5:5
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds W1 + W2 = W2 + W1 by Lm1;

theorem Th6: :: VECTSP_5:6
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M holds W1 + (W2 + W3) = (W1 + W2) + W3
proof end;

theorem Th7: :: VECTSP_5:7
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds
( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )
proof end;

theorem Th8: :: VECTSP_5:8
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 Th9: :: VECTSP_5:9
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being strict Subspace of M holds
( ((0). M) + W = W & W + ((0). M) = W )
proof end;

Lm4: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being Subspace of M ex W9 being strict Subspace of M st the carrier of W = the carrier of W9

proof end;

Lm5: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W, W9, W1 being Subspace of M st the carrier of W = the carrier of W9 holds
( W1 + W = W1 + W9 & W + W1 = W9 + W1 )

proof end;

Lm6:
proof end;

theorem :: VECTSP_5:10
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF holds
( ((0). M) + () = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & () + ((0). M) = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) ) by Th9;

theorem Th11: :: VECTSP_5:11
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being Subspace of M holds
( () + W = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & W + () = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) )
proof end;

theorem :: VECTSP_5:12

theorem :: VECTSP_5:13
proof end;

theorem Th14: :: VECTSP_5:14
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof end;

Lm7: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ W2) c= the carrier of W1

proof end;

theorem Th15: :: VECTSP_5:15
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds
( W1 /\ W2 is Subspace of W1 & W1 /\ W2 is Subspace of W2 )
proof end;

Lm8: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W, W9, W1 being Subspace of M st the carrier of W = the carrier of W9 holds
( W1 /\ W = W1 /\ W9 & W /\ W1 = W9 /\ W1 )

proof end;

theorem Th16: :: VECTSP_5:16
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:17
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
W1 /\ W3 is Subspace of W2 /\ W3
proof end;

theorem :: VECTSP_5:18
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W3 holds
W1 /\ W2 is Subspace of W3
proof end;

theorem :: VECTSP_5:19
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 Th20: :: VECTSP_5:20
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being Subspace of M holds
( ((0). M) /\ W = (0). M & W /\ ((0). M) = (0). M )
proof end;

theorem Th21: :: VECTSP_5:21
proof end;

theorem :: VECTSP_5:22

Lm9: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)

proof end;

theorem :: VECTSP_5:23
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds W1 /\ W2 is Subspace of W1 + W2 by ;

Lm10: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2

proof end;

theorem :: VECTSP_5:24
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1 being Subspace of M
for W2 being strict Subspace of M holds (W1 /\ W2) + W2 = W2 by ;

Lm11: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1

proof end;

theorem :: VECTSP_5:25
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W2 being Subspace of M
for W1 being strict Subspace of M holds W1 /\ (W1 + W2) = W1 by ;

Lm12: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:26
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M holds (W1 /\ W2) + (W2 /\ W3) is Subspace of W2 /\ (W1 + W3) by ;

Lm13: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:27
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3) by ;

Lm14: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: VECTSP_5:28
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M holds W2 + (W1 /\ W3) is Subspace of (W1 + W2) /\ (W2 + W3) by ;

Lm15: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:29
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3) by ;

theorem Th30: :: VECTSP_5:30
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W2, W3 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_5:31
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being strict Subspace of M holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
proof end;

theorem :: VECTSP_5:32
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:33
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 being Subspace of M st W1 is Subspace of W2 holds
W1 is Subspace of W2 + W3
proof end;

theorem :: VECTSP_5:34
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2, W3 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_5:35
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;
func Subspaces M -> set means :Def3: :: VECTSP_5:def 3
for x being object holds
( x in it iff ex W being strict Subspace of M st W = x );
existence
ex b1 being set st
for x being object holds
( x in b1 iff ex W being strict Subspace of M st W = x )
proof end;
uniqueness
for b1, b2 being set st ( for x being object holds
( x in b1 iff ex W being strict Subspace of M st W = x ) ) & ( for x being object holds
( x in b2 iff ex W being strict Subspace of M st W = x ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines Subspaces VECTSP_5:def 3 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for b3 being set holds
( b3 = Subspaces M iff for x being object holds
( x in b3 iff ex W being strict Subspace of M st W = x ) );

registration end;

theorem :: VECTSP_5:36
proof end;

definition
let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;
let W1, W2 be Subspace of M;
pred M is_the_direct_sum_of W1,W2 means :: VECTSP_5:def 4
( ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) = W1 + W2 & W1 /\ W2 = (0). M );
end;

:: deftheorem defines is_the_direct_sum_of VECTSP_5:def 4 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds
( M is_the_direct_sum_of W1,W2 iff ( ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) = W1 + W2 & W1 /\ W2 = (0). M ) );

Lm16: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds
( W1 + W2 = ModuleStr(# the carrier of M, the addF of M, the ZeroF 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
ex b1 being Subspace of V st V is_the_direct_sum_of b1,W
proof end;
end;

:: deftheorem Def5 defines Linear_Compl VECTSP_5:def 5 :
for F being Field
for V being VectSp of F
for W, b4 being Subspace of V holds
( b4 is Linear_Compl of W iff V is_the_direct_sum_of b4,W );

Lm17:
by Lm1;

theorem :: VECTSP_5:37
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 by ;

theorem Th38: :: VECTSP_5:38
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 ) by ;

theorem Th39: :: VECTSP_5:39
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 = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) & L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) )
proof end;

theorem Th40: :: VECTSP_5:40
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_5:41
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 Th42: :: VECTSP_5:42

theorem :: VECTSP_5:43
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_5:44
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 ) by ;

theorem Th45: :: VECTSP_5:45
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 Th46: :: VECTSP_5:46
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:47
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 Th48: :: VECTSP_5:48
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:49
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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
ex b1 being Element of [: the carrier of M, the carrier of M:] st
( v = (b1 1) + (b1 2) & b1 1 in W1 & b1 2 in W2 )
proof end;
uniqueness
for b1, b2 being Element of [: the carrier of M, the carrier of M:] st v = (b1 1) + (b1 2) & b1 1 in W1 & b1 2 in W2 & v = (b2 1) + (b2 2) & b2 1 in W1 & b2 2 in W2 holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines |-- VECTSP_5:def 6 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v being Element of M
for W1, W2 being Subspace of M st M is_the_direct_sum_of W1,W2 holds
for b6 being Element of [: the carrier of M, the carrier of M:] holds
( b6 = v |-- (W1,W2) iff ( v = (b6 1) + (b6 2) & b6 1 in W1 & b6 2 in W2 ) );

theorem Th50: :: VECTSP_5:50
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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 Th51: :: VECTSP_5:51
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over 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_5:52
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_5:53
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_5:54
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_5:55
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 by ;

theorem :: VECTSP_5:56
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 by ;

definition
let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;
func SubJoin M -> BinOp of () means :Def7: :: VECTSP_5:def 7
for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
it . (A1,A2) = W1 + W2;
existence
ex b1 being BinOp of () st
for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2
proof end;
uniqueness
for b1, b2 being BinOp of () st ( for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2 ) & ( for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 + W2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines SubJoin VECTSP_5:def 7 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for b3 being BinOp of () holds
( b3 = SubJoin M iff for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b3 . (A1,A2) = W1 + W2 );

definition
let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;
func SubMeet M -> BinOp of () means :Def8: :: VECTSP_5:def 8
for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
it . (A1,A2) = W1 /\ W2;
existence
ex b1 being BinOp of () st
for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2
proof end;
uniqueness
for b1, b2 being BinOp of () st ( for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2 ) & ( for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 /\ W2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines SubMeet VECTSP_5:def 8 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for b3 being BinOp of () holds
( b3 = SubMeet M iff for A1, A2 being Element of Subspaces M
for W1, W2 being Subspace of M st A1 = W1 & A2 = W2 holds
b3 . (A1,A2) = W1 /\ W2 );

theorem Th57: :: VECTSP_5:57
proof end;

theorem Th58: :: VECTSP_5:58
proof end;

theorem Th59: :: VECTSP_5:59
proof end;

theorem Th60: :: VECTSP_5:60
proof end;

theorem :: VECTSP_5:61
proof end;

theorem :: VECTSP_5:62
for F being Field
for V being VectSp of F holds LattStr(# (),(),() #) is C_Lattice
proof end;