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Theorem omllaw3 28565
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 21940 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 5765 . . . . . 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 28560 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OL )
4 omllaw3.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
5 eqid 2256 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
6 omllaw3.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
74, 5, 6olj01 28545 . . . . . . . 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 2289 . . . 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 28563 . . . . 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 2291 . 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 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   occoc 13143   joincjn 14005   meetcmee 14006   0.cp0 14070   OLcol 28494   OMLcoml 28495
This theorem is referenced by:  omlfh1N  28578  atlatmstc  28639
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-lub 14035  df-glb 14036  df-join 14037  df-p0 14072  df-lat 14079  df-oposet 28496  df-ol 28498  df-oml 28499
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