:: Operations on Subspaces in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received September 20, 1989
:: Copyright (c) 1990 Association of Mizar Users
:: deftheorem Def1 defines + RLSUB_2:def 1 :
:: deftheorem Def2 defines /\ RLSUB_2:def 2 :
theorem :: RLSUB_2:1
canceled;
theorem :: RLSUB_2:2
canceled;
theorem :: RLSUB_2:3
canceled;
theorem :: RLSUB_2:4
canceled;
theorem Th5: :: RLSUB_2:5
theorem Th6: :: RLSUB_2:6
theorem Th7: :: RLSUB_2:7
Lm1:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1
Lm2:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of W1 c= the carrier of (W1 + W2)
Lm3:
for V being RealLinearSpace
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 :: RLSUB_2:8
theorem :: RLSUB_2:9
theorem Th10: :: RLSUB_2:10
theorem Th11: :: RLSUB_2:11
theorem Th12: :: RLSUB_2:12
theorem Th13: :: RLSUB_2:13
theorem :: RLSUB_2:14
theorem Th15: :: RLSUB_2:15
theorem :: RLSUB_2:16
theorem :: RLSUB_2:17
theorem Th18: :: RLSUB_2:18
theorem Th19: :: RLSUB_2:19
Lm4:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of W1
theorem Th20: :: RLSUB_2:20
theorem Th21: :: RLSUB_2:21
theorem Th22: :: RLSUB_2:22
theorem :: RLSUB_2:23
theorem Th24: :: RLSUB_2:24
theorem :: RLSUB_2:25
Lm5:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
theorem :: RLSUB_2:26
Lm6:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
theorem :: RLSUB_2:27
Lm7:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
theorem :: RLSUB_2:28
Lm8:
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
theorem :: RLSUB_2:29
Lm9:
for V being RealLinearSpace
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 :: RLSUB_2:30
Lm10:
for V being RealLinearSpace
for W2, W1, W3 being Subspace of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
theorem :: RLSUB_2:31
Lm11:
for V being RealLinearSpace
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 :: RLSUB_2:32
theorem Th33: :: RLSUB_2:33
theorem :: RLSUB_2:34
theorem :: RLSUB_2:35
theorem :: RLSUB_2:36
:: deftheorem Def3 defines Subspaces RLSUB_2:def 3 :
theorem :: RLSUB_2:37
canceled;
theorem :: RLSUB_2:38
canceled;
theorem :: RLSUB_2:39
:: deftheorem Def4 defines is_the_direct_sum_of RLSUB_2:def 4 :
Lm12:
for V being RealLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = RLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #)
Lm13:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 + W2 = RLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult 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 non empty right_complementable add-associative right_zeroed addLoopStr
for v, v1, v2 being Element of V holds
( v = v1 + v2 iff v1 = v - v2 )
Lm15:
for V being RealLinearSpace
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 RLSUB_2:def 5 :
Lm16:
for V being RealLinearSpace
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 :: RLSUB_2:40
canceled;
theorem :: RLSUB_2:41
canceled;
theorem :: RLSUB_2:42
theorem Th43: :: RLSUB_2:43
theorem Th44: :: RLSUB_2:44
theorem Th45: :: RLSUB_2:45
theorem :: RLSUB_2:46
theorem Th47: :: RLSUB_2:47
theorem :: RLSUB_2:48
theorem :: RLSUB_2:49
theorem Th50: :: RLSUB_2:50
Lm17:
for V being RealLinearSpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C
theorem Th51: :: RLSUB_2:51
theorem :: RLSUB_2:52
theorem Th53: :: RLSUB_2:53
theorem :: RLSUB_2:54
:: deftheorem Def6 defines |-- RLSUB_2:def 6 :
theorem :: RLSUB_2:55
canceled;
theorem :: RLSUB_2:56
canceled;
theorem :: RLSUB_2:57
canceled;
theorem :: RLSUB_2:58
canceled;
theorem Th59: :: RLSUB_2:59
theorem Th60: :: RLSUB_2:60
theorem :: RLSUB_2:61
theorem :: RLSUB_2:62
theorem :: RLSUB_2:63
theorem :: RLSUB_2:64
theorem :: RLSUB_2:65
definition
let V be
RealLinearSpace;
func SubJoin V -> BinOp of
Subspaces V means :
Def7:
:: RLSUB_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 RLSUB_2:def 7 :
definition
let V be
RealLinearSpace;
func SubMeet V -> BinOp of
Subspaces V means :
Def8:
:: RLSUB_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 RLSUB_2:def 8 :
theorem :: RLSUB_2:66
canceled;
theorem :: RLSUB_2:67
canceled;
theorem :: RLSUB_2:68
canceled;
theorem :: 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
Lm18:
for l being Lattice
for a, b being Element of l holds
( a is_a_complement_of b iff ( a "\/" b = Top l & a "/\" b = Bottom l ) )
Lm19:
for l being Lattice
for b being Element of l st ( for a being Element of l holds a "/\" b = b ) holds
b = Bottom l
Lm20:
for l being Lattice
for b being Element of l st ( for a being Element of l holds a "\/" b = b ) holds
b = Top l
theorem Th75: :: RLSUB_2:75
registration
let V be
RealLinearSpace;
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 Th71, Th72, Th74, Th75;
end;
theorem :: RLSUB_2:76
theorem :: RLSUB_2:77
canceled;
theorem :: RLSUB_2:78
theorem :: RLSUB_2:79
theorem :: RLSUB_2:80
theorem :: RLSUB_2:81
canceled;
theorem :: RLSUB_2:82
canceled;
theorem :: RLSUB_2:83
canceled;
theorem :: RLSUB_2:84
theorem :: RLSUB_2:85