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Theorem cmt2N 29440
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22172 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 29432 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 976 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 cmt2.b . . . . . . 7  |-  B  =  ( Base `  K
)
4 eqid 2283 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
53, 4latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
61, 5syl3an1 1215 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
7 simp2 956 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 omlop 29431 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
983ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
10 simp3 957 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 cmt2.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
123, 11opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
139, 10, 12syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
143, 4latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)
152, 7, 13, 14syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  Y ) )  e.  B )
16 eqid 2283 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
173, 16latjcom 14165 . . . . 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 1182 . . . 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 29385 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
209, 10, 19syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
2120oveq2d 5874 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) )  =  ( X ( meet `  K
) Y ) )
2221oveq2d 5874 . . . 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 2318 . . 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 2294 . 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 29401 . 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 29401 . . 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 1227 . 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 276 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   cmccmtN 29363   OMLcoml 29365
This theorem is referenced by:  cmt3N  29441  cmt4N  29442  omlfh1N  29448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-join 14110  df-lat 14152  df-oposet 29366  df-cmtN 29367  df-ol 29368  df-oml 29369
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