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Theorem cmt2N 29416
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22936 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 29408 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 978 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 cmt2.b . . . . . . 7  |-  B  =  ( Base `  K
)
4 eqid 2380 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
53, 4latmcl 14400 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
61, 5syl3an1 1217 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
7 simp2 958 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 omlop 29407 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
983ad2ant1 978 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
10 simp3 959 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 cmt2.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
123, 11opoccl 29360 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
139, 10, 12syl2anc 643 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
143, 4latmcl 14400 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)
152, 7, 13, 14syl3anc 1184 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  Y ) )  e.  B )
16 eqid 2380 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
173, 16latjcom 14408 . . . . 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 1184 . . . 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 29361 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
209, 10, 19syl2anc 643 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
2120oveq2d 6029 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) )  =  ( X ( meet `  K
) Y ) )
2221oveq2d 6029 . . . 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 2415 . . 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 2391 . 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 29377 . 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 29377 . . 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 1229 . 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 277 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 177    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   occoc 13457   joincjn 14321   meetcmee 14322   Latclat 14394   OPcops 29338   cmccmtN 29339   OMLcoml 29341
This theorem is referenced by:  cmt3N  29417  cmt4N  29418  omlfh1N  29424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-join 14353  df-lat 14395  df-oposet 29342  df-cmtN 29343  df-ol 29344  df-oml 29345
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