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Theorem omllaw3 28702
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 22007 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 5827 . . . . . 6  |-  ( ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ->  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) )  =  ( X ( join `  K
)  .0.  ) )
21adantl 454 . . . . 5  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  )  -> 
( X ( join `  K ) ( Y 
./\  (  ._|_  `  X
) ) )  =  ( X ( join `  K )  .0.  )
)
3 omlol 28697 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OL )
4 omllaw3.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
5 eqid 2284 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
6 omllaw3.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
74, 5, 6olj01 28682 . . . . . . . 8  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X ( join `  K )  .0.  )  =  X )
83, 7sylan 459 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  ( X ( join `  K )  .0.  )  =  X )
983adant3 980 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( join `  K )  .0.  )  =  X )
109adantr 453 . . . . 5  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  )  -> 
( X ( join `  K )  .0.  )  =  X )
112, 10eqtr2d 2317 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  )  ->  X  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) )
1211adantrl 699 . . 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 28700 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) ) )
1716imp 420 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  Y  =  ( X ( join `  K
) ( Y  ./\  (  ._|_  `  X )
) ) )
1817adantrr 700 . . 3  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ) )  ->  Y  =  ( X
( join `  K )
( Y  ./\  (  ._|_  `  X ) ) ) )
1912, 18eqtr4d 2319 . 2  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .<_  Y  /\  ( Y  ./\  (  ._|_  `  X ) )  =  .0.  ) )  ->  X  =  Y )
2019ex 425 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 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   occoc 13210   joincjn 14072   meetcmee 14073   0.cp0 14137   OLcol 28631   OMLcoml 28632
This theorem is referenced by:  omlfh1N  28715  atlatmstc  28776
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  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 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  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 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-lub 14102  df-glb 14103  df-join 14104  df-p0 14139  df-lat 14146  df-oposet 28633  df-ol 28635  df-oml 28636
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