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Theorem omllaw2N 28713
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 22160 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 28712 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
7 eqcom 2286 . . 3  |-  ( ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  =  Y  <-> 
Y  =  ( X 
.\/  ( (  ._|_  `  X )  ./\  Y
) ) )
8 omllat 28711 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
983ad2ant1 976 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
10 omlop 28710 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
111, 5opoccl 28663 . . . . . . . 8  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
1210, 11sylan 457 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
13123adant3 975 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  X )  e.  B )
14 simp3 957 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
151, 4latmcom 14177 . . . . . 6  |-  ( ( K  e.  Lat  /\  (  ._|_  `  X )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  X )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X ) ) )
169, 13, 14, 15syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X )
) )
1716oveq2d 5836 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X
) ) ) )
1817eqeq2d 2295 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  =  ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
197, 18syl5bb 248 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  ( (  ._|_  `  X
)  ./\  Y )
)  =  Y  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
206, 19sylibrd 225 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   occoc 13212   joincjn 14074   meetcmee 14075   Latclat 14147   OPcops 28641   OMLcoml 28644
This theorem is referenced by:  omllaw5N  28716  cmtcomlemN  28717  cmtbr3N  28723
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-meet 14107  df-lat 14148  df-oposet 28645  df-ol 28647  df-oml 28648
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