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Theorem omllaw3 28586
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 21961 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 5786 . . . . . 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 28581 . . . . . . . 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 28566 . . . . . . . 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 28584 . . . . 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 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   occoc 13164   joincjn 14026   meetcmee 14027   0.cp0 14091   OLcol 28515   OMLcoml 28516
This theorem is referenced by:  omlfh1N  28599  atlatmstc  28660
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-lub 14056  df-glb 14057  df-join 14058  df-p0 14093  df-lat 14100  df-oposet 28517  df-ol 28519  df-oml 28520
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