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Theorem omllaw2N 34692
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 26372 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw.b  |-  B  =  ( Base `  K
)
omllaw.l  |-  .<_  =  ( le `  K )
omllaw.j  |-  .\/  =  ( join `  K )
omllaw.m  |-  ./\  =  ( meet `  K )
omllaw.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
omllaw2N  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  Y ) )

Proof of Theorem omllaw2N
StepHypRef Expression
1 omllaw.b . . 3  |-  B  =  ( Base `  K
)
2 omllaw.l . . 3  |-  .<_  =  ( le `  K )
3 omllaw.j . . 3  |-  .\/  =  ( join `  K )
4 omllaw.m . . 3  |-  ./\  =  ( meet `  K )
5 omllaw.o . . 3  |-  ._|_  =  ( oc `  K )
61, 2, 3, 4, 5omllaw 34691 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
7 eqcom 2450 . . 3  |-  ( ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  =  Y  <-> 
Y  =  ( X 
.\/  ( (  ._|_  `  X )  ./\  Y
) ) )
8 omllat 34690 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
983ad2ant1 1016 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
10 omlop 34689 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
111, 5opoccl 34642 . . . . . . . 8  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
1210, 11sylan 471 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
13123adant3 1015 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  X )  e.  B )
14 simp3 997 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
151, 4latmcom 15576 . . . . . 6  |-  ( ( K  e.  Lat  /\  (  ._|_  `  X )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  X )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X ) ) )
169, 13, 14, 15syl3anc 1227 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X )
) )
1716oveq2d 6294 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X
) ) ) )
1817eqeq2d 2455 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  =  ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
197, 18syl5bb 257 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  ( (  ._|_  `  X
)  ./\  Y )
)  =  Y  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
206, 19sylibrd 234 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1381    e. wcel 1802   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   Basecbs 14506   lecple 14578   occoc 14579   joincjn 15444   meetcmee 15445   Latclat 15546   OPcops 34620   OMLcoml 34623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-glb 15476  df-meet 15478  df-lat 15547  df-oposet 34624  df-ol 34626  df-oml 34627
This theorem is referenced by:  omllaw5N  34695  cmtcomlemN  34696  cmtbr3N  34702
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