begin
:: deftheorem Def1 defines + VECTSP_5:def 1 :
Lm1:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds W1 + W2 = W2 + W1
:: deftheorem Def2 defines /\ VECTSP_5:def 2 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th5:
theorem Th6:
theorem Th7:
Lm2:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of W1 c= the carrier of (W1 + W2)
Lm3:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed 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
theorem
theorem
theorem Th10:
theorem Th11:
theorem Th12:
theorem Th13:
Lm4:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W being Subspace of M ex W9 being strict Subspace of M st the carrier of W = the carrier of W9
Lm5:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of 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 )
Lm6:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W being Subspace of M holds W is Subspace of (Omega). M
theorem
theorem Th15:
theorem
theorem
theorem
canceled;
theorem Th19:
Lm7:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ W2) c= the carrier of W1
theorem Th20:
Lm8:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of 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 )
theorem Th21:
theorem
theorem
theorem
theorem Th25:
theorem
canceled;
theorem Th27:
theorem
Lm9:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
theorem
Lm10:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
theorem
Lm11:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
theorem
Lm12:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed 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))
theorem
Lm13:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed 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))
theorem
Lm14:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed 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))
theorem
Lm15:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed 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))
theorem
theorem Th36:
theorem
theorem
theorem
theorem
theorem
:: deftheorem Def3 defines Subspaces VECTSP_5:def 3 :
theorem
canceled;
theorem
canceled;
theorem
:: deftheorem Def4 defines is_the_direct_sum_of VECTSP_5:def 4 :
Lm16:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds
( W1 + W2 = VectSpStr(# 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 ) )
:: deftheorem Def5 defines Linear_Compl VECTSP_5:def 5 :
Lm17:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed 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
theorem
canceled;
theorem
canceled;
theorem
theorem Th48:
theorem Th49:
theorem Th50:
theorem
theorem Th52:
theorem
theorem
theorem Th55:
theorem Th56:
theorem
theorem Th58:
theorem
:: deftheorem Def6 defines |-- VECTSP_5:def 6 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th64:
theorem Th65:
theorem
theorem
theorem
theorem
theorem
definition
let GF be non
empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ;
let M be non
empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of
GF;
func SubJoin M -> BinOp of
(Subspaces M) means :
Def7:
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 (Subspaces M) 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
uniqueness
for b1, b2 being BinOp of (Subspaces M) 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
end;
:: deftheorem Def7 defines SubJoin VECTSP_5:def 7 :
definition
let GF be non
empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ;
let M be non
empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of
GF;
func SubMeet M -> BinOp of
(Subspaces M) means :
Def8:
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 (Subspaces M) 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
uniqueness
for b1, b2 being BinOp of (Subspaces M) 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
end;
:: deftheorem Def8 defines SubMeet VECTSP_5:def 8 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th75:
theorem Th76:
theorem Th77:
theorem Th78:
theorem
theorem