:: Operations on Subspaces in Vector Space
:: by Wojciech A. Trybulec
::
:: Received July 27, 1990
:: Copyright (c) 1990 Association of Mizar Users
:: 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 VectSp-like 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 :: VECTSP_5:1
canceled;
theorem :: VECTSP_5:2
canceled;
theorem :: VECTSP_5:3
canceled;
theorem :: VECTSP_5:4
canceled;
theorem Th5: :: VECTSP_5:5
theorem Th6: :: VECTSP_5:6
theorem Th7: :: VECTSP_5:7
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 VectSp-like 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 VectSp-like 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 :: VECTSP_5:8
theorem :: VECTSP_5:9
theorem Th10: :: VECTSP_5:10
theorem Th11: :: VECTSP_5:11
theorem Th12: :: VECTSP_5:12
theorem Th13: :: VECTSP_5:13
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 VectSp-like Abelian add-associative right_zeroed 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'
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 VectSp-like Abelian add-associative right_zeroed 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 )
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 VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for W being Subspace of M holds W is Subspace of (Omega). M
theorem :: VECTSP_5:14
theorem Th15: :: VECTSP_5:15
theorem :: VECTSP_5:16
theorem :: VECTSP_5:17
theorem :: VECTSP_5:18
canceled;
theorem Th19: :: VECTSP_5:19
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 VectSp-like 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: :: VECTSP_5:20
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 VectSp-like Abelian add-associative right_zeroed 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 )
theorem Th21: :: VECTSP_5:21
theorem :: VECTSP_5:22
theorem :: VECTSP_5:23
theorem :: VECTSP_5:24
theorem Th25: :: VECTSP_5:25
theorem :: VECTSP_5:26
canceled;
theorem Th27: :: VECTSP_5:27
theorem :: VECTSP_5:28
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 VectSp-like 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 :: VECTSP_5:29
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 VectSp-like 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 :: VECTSP_5:30
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 VectSp-like 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 :: VECTSP_5:31
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 VectSp-like 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 :: VECTSP_5:32
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 VectSp-like 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 :: VECTSP_5:33
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 VectSp-like 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 :: VECTSP_5:34
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 VectSp-like 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 :: VECTSP_5:35
theorem Th36: :: VECTSP_5:36
theorem :: VECTSP_5:37
theorem :: VECTSP_5:38
theorem :: VECTSP_5:39
theorem :: VECTSP_5:40
theorem :: VECTSP_5:41
:: deftheorem Def3 defines Subspaces VECTSP_5:def 3 :
theorem :: VECTSP_5:42
canceled;
theorem :: VECTSP_5:43
canceled;
theorem :: VECTSP_5:44
:: 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 VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds
( W1 + W2 = VectSpStr(# the carrier of M,the U7 of M,the U2 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 VectSp-like 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 :: VECTSP_5:45
canceled;
theorem :: VECTSP_5:46
canceled;
theorem :: VECTSP_5:47
theorem Th48: :: VECTSP_5:48
theorem Th49: :: VECTSP_5:49
theorem Th50: :: VECTSP_5:50
theorem :: VECTSP_5:51
theorem Th52: :: VECTSP_5:52
theorem :: VECTSP_5:53
theorem :: VECTSP_5:54
theorem Th55: :: VECTSP_5:55
theorem Th56: :: VECTSP_5:56
theorem :: VECTSP_5:57
theorem Th58: :: VECTSP_5:58
theorem :: VECTSP_5:59
:: 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
theorem Th65: :: VECTSP_5:65
theorem :: VECTSP_5:66
theorem :: VECTSP_5:67
theorem :: VECTSP_5:68
theorem :: VECTSP_5:69
theorem :: VECTSP_5:70
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 VectSp-like Abelian add-associative right_zeroed VectSpStr of
GF;
func SubJoin M -> BinOp of
Subspaces M 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 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 VectSp-like Abelian add-associative right_zeroed VectSpStr of
GF;
func SubMeet M -> BinOp of
Subspaces M 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 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 :: VECTSP_5:71
canceled;
theorem :: VECTSP_5:72
canceled;
theorem :: VECTSP_5:73
canceled;
theorem :: VECTSP_5:74
canceled;
theorem Th75: :: VECTSP_5:75
theorem Th76: :: VECTSP_5:76
theorem Th77: :: VECTSP_5:77
theorem Th78: :: VECTSP_5:78
theorem :: VECTSP_5:79
theorem :: VECTSP_5:80