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Theorem cmt2N 30122
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 23100 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmt2.b  |-  B  =  ( Base `  K
)
cmt2.o  |-  ._|_  =  ( oc `  K )
cmt2.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
cmt2N  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X C (  ._|_  `  Y ) ) )

Proof of Theorem cmt2N
StepHypRef Expression
1 omllat 30114 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 979 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 cmt2.b . . . . . . 7  |-  B  =  ( Base `  K
)
4 eqid 2438 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
53, 4latmcl 14485 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
61, 5syl3an1 1218 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
7 simp2 959 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 omlop 30113 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
983ad2ant1 979 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
10 simp3 960 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 cmt2.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
123, 11opoccl 30066 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
139, 10, 12syl2anc 644 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
143, 4latmcl 14485 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)
152, 7, 13, 14syl3anc 1185 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  Y ) )  e.  B )
16 eqid 2438 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
173, 16latjcom 14493 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X ( meet `  K
) Y )  e.  B  /\  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)  ->  ( ( X ( meet `  K
) Y ) (
join `  K )
( X ( meet `  K ) (  ._|_  `  Y ) ) )  =  ( ( X ( meet `  K
) (  ._|_  `  Y
) ) ( join `  K ) ( X ( meet `  K
) Y ) ) )
182, 6, 15, 17syl3anc 1185 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K ) Y ) ( join `  K ) ( X ( meet `  K
) (  ._|_  `  Y
) ) )  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K ) Y ) ) )
193, 11opococ 30067 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
209, 10, 19syl2anc 644 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
2120oveq2d 6100 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) )  =  ( X ( meet `  K
) Y ) )
2221oveq2d 6100 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) )  =  ( ( X ( meet `  K
) (  ._|_  `  Y
) ) ( join `  K ) ( X ( meet `  K
) Y ) ) )
2318, 22eqtr4d 2473 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K ) Y ) ( join `  K ) ( X ( meet `  K
) (  ._|_  `  Y
) ) )  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) ) )
2423eqeq2d 2449 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( ( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  Y )
) )  <->  X  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) ) ) )
25 cmt2.c . . 3  |-  C  =  ( cm `  K
)
263, 16, 4, 11, 25cmtvalN 30083 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X  =  ( ( X ( meet `  K
) Y ) (
join `  K )
( X ( meet `  K ) (  ._|_  `  Y ) ) ) ) )
273, 16, 4, 11, 25cmtvalN 30083 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X C (  ._|_  `  Y
)  <->  X  =  (
( X ( meet `  K ) (  ._|_  `  Y ) ) (
join `  K )
( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) ) ) ) )
2813, 27syld3an3 1230 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( 
._|_  `  Y )  <->  X  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) ) ) )
2924, 26, 283bitr4d 278 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X C (  ._|_  `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   occoc 13542   joincjn 14406   meetcmee 14407   Latclat 14479   OPcops 30044   cmccmtN 30045   OMLcoml 30047
This theorem is referenced by:  cmt3N  30123  cmt4N  30124  omlfh1N  30130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-join 14438  df-lat 14480  df-oposet 30048  df-cmtN 30049  df-ol 30050  df-oml 30051
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