:: RUSUB_2 semantic presentation
:: deftheorem Def1 defines + RUSUB_2:def 1 :
:: deftheorem Def2 defines /\ RUSUB_2:def 2 :
theorem Th1: :: RUSUB_2:1
theorem Th2: :: RUSUB_2:2
theorem Th3: :: RUSUB_2:3
Lemma36:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1
Lemma39:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of W1 c= the carrier of (W1 + W2)
Lemma40:
for V being RealUnitarySpace
for W1 being Subspace of V
for W2 being strict Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
theorem Th4: :: RUSUB_2:4
theorem Th5: :: RUSUB_2:5
theorem Th6: :: RUSUB_2:6
theorem Th7: :: RUSUB_2:7
theorem Th8: :: RUSUB_2:8
theorem Th9: :: RUSUB_2:9
theorem Th10: :: RUSUB_2:10
theorem Th11: :: RUSUB_2:11
theorem Th12: :: RUSUB_2:12
theorem Th13: :: RUSUB_2:13
theorem Th14: :: RUSUB_2:14
theorem Th15: :: RUSUB_2:15
Lemma65:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of W1
theorem Th16: :: RUSUB_2:16
theorem Th17: :: RUSUB_2:17
theorem Th18: :: RUSUB_2:18
theorem Th19: :: RUSUB_2:19
theorem Th20: :: RUSUB_2:20
theorem Th21: :: RUSUB_2:21
Lemma70:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
theorem Th22: :: RUSUB_2:22
Lemma71:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
theorem Th23: :: RUSUB_2:23
Lemma73:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
theorem Th24: :: RUSUB_2:24
Lemma76:
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
theorem Th25: :: RUSUB_2:25
Lemma77:
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))
theorem Th26: :: RUSUB_2:26
Lemma79:
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
theorem Th27: :: RUSUB_2:27
Lemma80:
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
theorem Th28: :: RUSUB_2:28
theorem Th29: :: RUSUB_2:29
theorem Th30: :: RUSUB_2:30
theorem Th31: :: RUSUB_2:31
theorem Th32: :: RUSUB_2:32
:: deftheorem Def3 defines Subspaces RUSUB_2:def 3 :
theorem Th33: :: RUSUB_2:33
:: deftheorem Def4 defines is_the_direct_sum_of RUSUB_2:def 4 :
Lemma92:
for V being RealUnitarySpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = UNITSTR(# the carrier of V,the Zero of V,the add of V,the Mult of V,the scalar of V #)
Lemma93:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( W1 + W2 = UNITSTR(# the carrier of V,the Zero of V,the add of V,the Mult of V,the scalar of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
Lemma94:
for V being RealUnitarySpace
for W being Subspace of V ex C being strict Subspace of V st V is_the_direct_sum_of C,W
:: deftheorem Def5 defines Linear_Compl RUSUB_2:def 5 :
Lemma190:
for V being RealUnitarySpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
theorem Th34: :: RUSUB_2:34
theorem Th35: :: RUSUB_2:35
theorem Th36: :: RUSUB_2:36
theorem Th37: :: RUSUB_2:37
theorem Th38: :: RUSUB_2:38
theorem Th39: :: RUSUB_2:39
theorem Th40: :: RUSUB_2:40
theorem Th41: :: RUSUB_2:41
Lemma195:
for V being RealUnitarySpace
for W being Subspace of V
for v being VECTOR of V
for x being set holds
( x in v + W iff ex u being VECTOR of V st
( u in W & x = v + u ) )
theorem Th42: :: RUSUB_2:42
Lemma199:
for V being RealUnitarySpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C
theorem Th43: :: RUSUB_2:43
theorem Th44: :: RUSUB_2:44
theorem Th45: :: RUSUB_2:45
theorem Th46: :: RUSUB_2:46
:: deftheorem Def6 defines |-- RUSUB_2:def 6 :
theorem Th47: :: RUSUB_2:47
theorem Th48: :: RUSUB_2:48
theorem Th49: :: RUSUB_2:49
theorem Th50: :: RUSUB_2:50
theorem Th51: :: RUSUB_2:51
theorem Th52: :: RUSUB_2:52
theorem Th53: :: RUSUB_2:53
definition
let V be
RealUnitarySpace;
func SubJoin c1 -> BinOp of
Subspaces a1 means :
Def7:
:: RUSUB_2:def 7
for
A1,
A2 being
Element of
Subspaces V for
W1,
W2 being
Subspace of
V st
A1 = W1 &
A2 = W2 holds
it . A1,
A2 = W1 + W2;
existence
ex b1 being BinOp of Subspaces V st
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 + W2
uniqueness
for b1, b2 being BinOp of Subspaces V st ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 + W2 ) & ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b2 . A1,A2 = W1 + W2 ) holds
b1 = b2
end;
:: deftheorem Def7 defines SubJoin RUSUB_2:def 7 :
definition
let V be
RealUnitarySpace;
func SubMeet c1 -> BinOp of
Subspaces a1 means :
Def8:
:: RUSUB_2:def 8
for
A1,
A2 being
Element of
Subspaces V for
W1,
W2 being
Subspace of
V st
A1 = W1 &
A2 = W2 holds
it . A1,
A2 = W1 /\ W2;
existence
ex b1 being BinOp of Subspaces V st
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 /\ W2
uniqueness
for b1, b2 being BinOp of Subspaces V st ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 /\ W2 ) & ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b2 . A1,A2 = W1 /\ W2 ) holds
b1 = b2
end;
:: deftheorem Def8 defines SubMeet RUSUB_2:def 8 :
theorem Th54: :: RUSUB_2:54
theorem Th55: :: RUSUB_2:55
theorem Th56: :: RUSUB_2:56
theorem Th57: :: RUSUB_2:57
theorem Th58: :: RUSUB_2:58
theorem Th59: :: RUSUB_2:59
registration
let V be
RealUnitarySpace;
cluster LattStr(#
(Subspaces a1),
(SubJoin a1),
(SubMeet a1) #)
-> Lattice-like modular lower-bounded upper-bounded complemented ;
coherence
( LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is lower-bounded & LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded & LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is modular & LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is complemented )
by , , Th7, Th8;
end;
theorem Th60: :: RUSUB_2:60
theorem Th61: :: RUSUB_2:61
theorem Th62: :: RUSUB_2:62
theorem Th63: :: RUSUB_2:63