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Theorem omllaw3 29728
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 22891 analog.) (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
omllaw3.b  |-  B  =  ( Base `  K
)
omllaw3.l  |-  .<_  =  ( le `  K )
omllaw3.m  |-  ./\  =  ( meet `  K )
omllaw3.o  |-  ._|_  =  ( oc `  K )
omllaw3.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
omllaw3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X )
)  =  .0.  )  ->  X  =  Y ) )

Proof of Theorem omllaw3
StepHypRef Expression
1 oveq2 6048 . . . . . 6  |-  ( ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ->  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) )  =  ( X ( join `  K
)  .0.  ) )
21adantl 453 . . . . 5  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  )  -> 
( X ( join `  K ) ( Y 
./\  (  ._|_  `  X
) ) )  =  ( X ( join `  K )  .0.  )
)
3 omlol 29723 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OL )
4 omllaw3.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
5 eqid 2404 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
6 omllaw3.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
74, 5, 6olj01 29708 . . . . . . . 8  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X ( join `  K )  .0.  )  =  X )
83, 7sylan 458 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  ( X ( join `  K )  .0.  )  =  X )
983adant3 977 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( join `  K )  .0.  )  =  X )
109adantr 452 . . . . 5  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  )  -> 
( X ( join `  K )  .0.  )  =  X )
112, 10eqtr2d 2437 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  )  ->  X  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) )
1211adantrl 697 . . 3  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ) )  ->  X  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) )
13 omllaw3.l . . . . . 6  |-  .<_  =  ( le `  K )
14 omllaw3.m . . . . . 6  |-  ./\  =  ( meet `  K )
15 omllaw3.o . . . . . 6  |-  ._|_  =  ( oc `  K )
164, 13, 5, 14, 15omllaw 29726 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) ) )
1716imp 419 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  Y  =  ( X ( join `  K
) ( Y  ./\  (  ._|_  `  X )
) ) )
1817adantrr 698 . . 3  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ) )  ->  Y  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) )
1912, 18eqtr4d 2439 . 2  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ) )  ->  X  =  Y )
2019ex 424 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X )
)  =  .0.  )  ->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   joincjn 14356   meetcmee 14357   0.cp0 14421   OLcol 29657   OMLcoml 29658
This theorem is referenced by:  omlfh1N  29741  atlatmstc  29802
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-p0 14423  df-lat 14430  df-oposet 29659  df-ol 29661  df-oml 29662
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