:: RLSUB_2 semantic presentation
:: deftheorem Def1 defines + RLSUB_2:def 1 :
:: deftheorem Def2 defines /\ RLSUB_2:def 2 :
theorem Th1: :: RLSUB_2:1
canceled;
theorem Th2: :: RLSUB_2:2
canceled;
theorem Th3: :: RLSUB_2:3
canceled;
theorem Th4: :: RLSUB_2:4
canceled;
theorem Th5: :: RLSUB_2:5
theorem Th6: :: RLSUB_2:6
theorem Th7: :: RLSUB_2:7
Lemma6:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds b2 + b3 = b3 + b2
Lemma7:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds the carrier of b2 c= the carrier of (b2 + b3)
Lemma8:
for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
( the carrier of b2 c= the carrier of b3 implies b2 + b3 = b3 )
theorem Th8: :: RLSUB_2:8
theorem Th9: :: RLSUB_2:9
theorem Th10: :: RLSUB_2:10
theorem Th11: :: RLSUB_2:11
theorem Th12: :: RLSUB_2:12
theorem Th13: :: RLSUB_2:13
theorem Th14: :: RLSUB_2:14
theorem Th15: :: RLSUB_2:15
theorem Th16: :: RLSUB_2:16
theorem Th17: :: RLSUB_2:17
theorem Th18: :: RLSUB_2:18
theorem Th19: :: RLSUB_2:19
Lemma17:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds the carrier of (b2 /\ b3) c= the carrier of b2
theorem Th20: :: RLSUB_2:20
theorem Th21: :: RLSUB_2:21
theorem Th22: :: RLSUB_2:22
theorem Th23: :: RLSUB_2:23
theorem Th24: :: RLSUB_2:24
theorem Th25: :: RLSUB_2:25
Lemma22:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds the carrier of (b2 /\ b3) c= the carrier of (b2 + b3)
theorem Th26: :: RLSUB_2:26
Lemma23:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds the carrier of ((b2 /\ b3) + b3) = the carrier of b3
theorem Th27: :: RLSUB_2:27
Lemma25:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds the carrier of (b2 /\ (b2 + b3)) = the carrier of b2
theorem Th28: :: RLSUB_2:28
Lemma27:
for b1 being RealLinearSpace
for b2, b3, b4 being Subspace of b1 holds the carrier of ((b2 /\ b3) + (b3 /\ b4)) c= the carrier of (b3 /\ (b2 + b4))
theorem Th29: :: RLSUB_2:29
Lemma28:
for b1 being RealLinearSpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b3 implies the carrier of (b3 /\ (b2 + b4)) = the carrier of ((b2 /\ b3) + (b3 /\ b4)) )
theorem Th30: :: RLSUB_2:30
Lemma30:
for b1 being RealLinearSpace
for b2, b3, b4 being Subspace of b1 holds the carrier of (b2 + (b3 /\ b4)) c= the carrier of ((b3 + b2) /\ (b2 + b4))
theorem Th31: :: RLSUB_2:31
Lemma31:
for b1 being RealLinearSpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b3 implies the carrier of (b3 + (b2 /\ b4)) = the carrier of ((b2 + b3) /\ (b3 + b4)) )
theorem Th32: :: RLSUB_2:32
theorem Th33: :: RLSUB_2:33
theorem Th34: :: RLSUB_2:34
theorem Th35: :: RLSUB_2:35
theorem Th36: :: RLSUB_2:36
:: deftheorem Def3 defines Subspaces RLSUB_2:def 3 :
theorem Th37: :: RLSUB_2:37
canceled;
theorem Th38: :: RLSUB_2:38
canceled;
theorem Th39: :: RLSUB_2:39
:: deftheorem Def4 defines is_the_direct_sum_of RLSUB_2:def 4 :
Lemma35:
for b1 being RealLinearSpace
for b2 being strict Subspace of b1 holds
( ( for b3 being VECTOR of b1 holds b3 in b2 ) implies b2 = RLSStruct(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1 #) )
Lemma36:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b2 + b3 = RLSStruct(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1 #) iff for b4 being VECTOR of b1 holds
ex b5, b6 being VECTOR of b1 st
( b5 in b2 & b6 in b3 & b4 = b5 + b6 ) )
Lemma37:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3, b4 being Element of b1 holds
( b2 = b3 + b4 iff b3 = b2 - b4 )
Lemma38:
for b1 being RealLinearSpace
for b2 being Subspace of b1 holds
ex b3 being strict Subspace of b1 st b1 is_the_direct_sum_of b3,b2
:: deftheorem Def5 defines Linear_Compl RLSUB_2:def 5 :
Lemma40:
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies b1 is_the_direct_sum_of b3,b2 )
theorem Th40: :: RLSUB_2:40
canceled;
theorem Th41: :: RLSUB_2:41
canceled;
theorem Th42: :: RLSUB_2:42
theorem Th43: :: RLSUB_2:43
theorem Th44: :: RLSUB_2:44
theorem Th45: :: RLSUB_2:45
theorem Th46: :: RLSUB_2:46
theorem Th47: :: RLSUB_2:47
theorem Th48: :: RLSUB_2:48
theorem Th49: :: RLSUB_2:49
theorem Th50: :: RLSUB_2:50
Lemma46:
for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being VECTOR of b1 holds
ex b4 being Coset of b2 st b3 in b4
theorem Th51: :: RLSUB_2:51
theorem Th52: :: RLSUB_2:52
theorem Th53: :: RLSUB_2:53
Lemma49:
for b1, b2 being set holds
not ( b1 c< b2 & ( for b3 being set holds
not ( b3 in b2 & not b3 in b1 ) ) )
theorem Th54: :: RLSUB_2:54
:: deftheorem Def6 defines |-- RLSUB_2:def 6 :
theorem Th55: :: RLSUB_2:55
canceled;
theorem Th56: :: RLSUB_2:56
canceled;
theorem Th57: :: RLSUB_2:57
canceled;
theorem Th58: :: RLSUB_2:58
canceled;
theorem Th59: :: RLSUB_2:59
theorem Th60: :: RLSUB_2:60
theorem Th61: :: RLSUB_2:61
theorem Th62: :: RLSUB_2:62
theorem Th63: :: RLSUB_2:63
theorem Th64: :: RLSUB_2:64
theorem Th65: :: RLSUB_2:65
definition
let c
1 be
RealLinearSpace;
func SubJoin c
1 -> BinOp of
Subspaces a
1 means :
Def7:
:: RLSUB_2:def 7
for b
1, b
2 being
Element of
Subspaces a
1for b
3, b
4 being
Subspace of a
1 holds
( b
1 = b
3 & b
2 = b
4 implies a
2 . b
1,b
2 = b
3 + b
4 );
existence
ex b1 being BinOp of Subspaces c1 st
for b2, b3 being Element of Subspaces c1
for b4, b5 being Subspace of c1 holds
( b2 = b4 & b3 = b5 implies b1 . b2,b3 = b4 + b5 )
uniqueness
for b1, b2 being BinOp of Subspaces c1 holds
( ( for b3, b4 being Element of Subspaces c1
for b5, b6 being Subspace of c1 holds
( b3 = b5 & b4 = b6 implies b1 . b3,b4 = b5 + b6 ) ) & ( for b3, b4 being Element of Subspaces c1
for b5, b6 being Subspace of c1 holds
( b3 = b5 & b4 = b6 implies b2 . b3,b4 = b5 + b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines SubJoin RLSUB_2:def 7 :
definition
let c
1 be
RealLinearSpace;
func SubMeet c
1 -> BinOp of
Subspaces a
1 means :
Def8:
:: RLSUB_2:def 8
for b
1, b
2 being
Element of
Subspaces a
1for b
3, b
4 being
Subspace of a
1 holds
( b
1 = b
3 & b
2 = b
4 implies a
2 . b
1,b
2 = b
3 /\ b
4 );
existence
ex b1 being BinOp of Subspaces c1 st
for b2, b3 being Element of Subspaces c1
for b4, b5 being Subspace of c1 holds
( b2 = b4 & b3 = b5 implies b1 . b2,b3 = b4 /\ b5 )
uniqueness
for b1, b2 being BinOp of Subspaces c1 holds
( ( for b3, b4 being Element of Subspaces c1
for b5, b6 being Subspace of c1 holds
( b3 = b5 & b4 = b6 implies b1 . b3,b4 = b5 /\ b6 ) ) & ( for b3, b4 being Element of Subspaces c1
for b5, b6 being Subspace of c1 holds
( b3 = b5 & b4 = b6 implies b2 . b3,b4 = b5 /\ b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines SubMeet RLSUB_2:def 8 :
theorem Th66: :: RLSUB_2:66
canceled;
theorem Th67: :: RLSUB_2:67
canceled;
theorem Th68: :: RLSUB_2:68
canceled;
theorem Th69: :: RLSUB_2:69
canceled;
theorem Th70: :: RLSUB_2:70
theorem Th71: :: RLSUB_2:71
theorem Th72: :: RLSUB_2:72
theorem Th73: :: RLSUB_2:73
theorem Th74: :: RLSUB_2:74
Lemma60:
for b1 being Lattice
for b2, b3 being Element of b1 holds
( b2 is_a_complement_of b3 iff ( b2 "\/" b3 = Top b1 & b2 "/\" b3 = Bottom b1 ) )
Lemma61:
for b1 being Lattice
for b2 being Element of b1 holds
( ( for b3 being Element of b1 holds b3 "/\" b2 = b2 ) implies b2 = Bottom b1 )
Lemma62:
for b1 being Lattice
for b2 being Element of b1 holds
( ( for b3 being Element of b1 holds b3 "\/" b2 = b2 ) implies b2 = Top b1 )
theorem Th75: :: RLSUB_2:75
registration
let c
1 be
RealLinearSpace;
cluster LattStr(#
(Subspaces a1),
(SubJoin a1),
(SubMeet a1) #)
-> Lattice-like modular lower-bounded upper-bounded complemented ;
coherence
( LattStr(# (Subspaces c1),(SubJoin c1),(SubMeet c1) #) is lower-bounded & LattStr(# (Subspaces c1),(SubJoin c1),(SubMeet c1) #) is upper-bounded & LattStr(# (Subspaces c1),(SubJoin c1),(SubMeet c1) #) is modular & LattStr(# (Subspaces c1),(SubJoin c1),(SubMeet c1) #) is complemented )
by Th71, Th72, Th74, Th75;
end;
theorem Th76: :: RLSUB_2:76
theorem Th77: :: RLSUB_2:77
for b
1, b
2 being
set holds
not ( b
1 c< b
2 & ( for b
3 being
set holds
not ( b
3 in b
2 & not b
3 in b
1 ) ) )
by Lemma49;
theorem Th78: :: RLSUB_2:78
theorem Th79: :: RLSUB_2:79
theorem Th80: :: RLSUB_2:80
theorem Th81: :: RLSUB_2:81
canceled;
theorem Th82: :: RLSUB_2:82
canceled;
theorem Th83: :: RLSUB_2:83
canceled;
theorem Th84: :: RLSUB_2:84
theorem Th85: :: RLSUB_2:85