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Theorem omllaw2N 34442
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 26326 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 34441 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
7 eqcom 2476 . . 3  |-  ( ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  =  Y  <-> 
Y  =  ( X 
.\/  ( (  ._|_  `  X )  ./\  Y
) ) )
8 omllat 34440 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
983ad2ant1 1017 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
10 omlop 34439 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
111, 5opoccl 34392 . . . . . . . 8  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
1210, 11sylan 471 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
13123adant3 1016 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  X )  e.  B )
14 simp3 998 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
151, 4latmcom 15579 . . . . . 6  |-  ( ( K  e.  Lat  /\  (  ._|_  `  X )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  X )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X ) ) )
169, 13, 14, 15syl3anc 1228 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X )
) )
1716oveq2d 6311 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X
) ) ) )
1817eqeq2d 2481 . . 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   occoc 14580   joincjn 15448   meetcmee 15449   Latclat 15549   OPcops 34370   OMLcoml 34373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-glb 15479  df-meet 15481  df-lat 15550  df-oposet 34374  df-ol 34376  df-oml 34377
This theorem is referenced by:  omllaw5N  34445  cmtcomlemN  34446  cmtbr3N  34452
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