:: Operations on Subspaces in Real Unitary Space
:: by Noboru Endou , Takashi Mitsuishi and Yasunari Shidama
::
:: Received October 9, 2002
:: Copyright (c) 2002 Association of Mizar Users
:: 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
Lm1:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1
Lm2:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of W1 c= the carrier of (W1 + W2)
Lm3:
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 :: RUSUB_2:4
theorem :: 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 :: RUSUB_2:12
theorem :: RUSUB_2:13
theorem Th14: :: RUSUB_2:14
theorem Th15: :: RUSUB_2:15
Lm4:
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 :: RUSUB_2:19
theorem Th20: :: RUSUB_2:20
theorem :: RUSUB_2:21
Lm5:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
theorem :: RUSUB_2:22
Lm6:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
theorem :: RUSUB_2:23
Lm7:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
theorem :: RUSUB_2:24
Lm8:
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 :: RUSUB_2:25
Lm9:
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 :: RUSUB_2:26
Lm10:
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 :: RUSUB_2:27
Lm11:
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 :: RUSUB_2:28
theorem Th29: :: RUSUB_2:29
theorem :: RUSUB_2:30
theorem :: RUSUB_2:31
theorem :: RUSUB_2:32
:: deftheorem Def3 defines Subspaces RUSUB_2:def 3 :
theorem :: RUSUB_2:33
:: deftheorem Def4 defines is_the_direct_sum_of RUSUB_2:def 4 :
Lm12:
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 U2 of V,the U7 of V,the Mult of V,the scalar of V #)
Lm13:
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( W1 + W2 = UNITSTR(# the carrier of V,the U2 of V,the U7 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 ) )
Lm14:
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 :
Lm15:
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 :: RUSUB_2:34
theorem Th35: :: RUSUB_2:35
theorem Th36: :: RUSUB_2:36
theorem Th37: :: RUSUB_2:37
theorem :: RUSUB_2:38
theorem Th39: :: RUSUB_2:39
theorem :: RUSUB_2:40
theorem :: RUSUB_2:41
Lm16:
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
Lm17:
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 :: RUSUB_2:44
theorem Th45: :: RUSUB_2:45
theorem :: RUSUB_2:46
:: deftheorem Def6 defines |-- RUSUB_2:def 6 :
theorem Th47: :: RUSUB_2:47
theorem Th48: :: RUSUB_2:48
theorem :: RUSUB_2:49
theorem :: RUSUB_2:50
theorem :: RUSUB_2:51
theorem :: RUSUB_2:52
theorem :: RUSUB_2:53
definition
let V be
RealUnitarySpace;
func SubJoin V -> BinOp of
Subspaces V 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 V -> BinOp of
Subspaces V 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 V),
(SubJoin V),
(SubMeet V) #)
-> 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 Th55, Th56, Th58, Th59;
end;
theorem :: RUSUB_2:60
theorem :: RUSUB_2:61
theorem :: RUSUB_2:62
theorem :: RUSUB_2:63