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Theorem opnoncon 30006
Description: Law of contradiction for orthoposets. (chocin 22997 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opnoncon.b  |-  B  =  ( Base `  K
)
opnoncon.o  |-  ._|_  =  ( oc `  K )
opnoncon.m  |-  ./\  =  ( meet `  K )
opnoncon.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
opnoncon  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X ) )  =  .0.  )

Proof of Theorem opnoncon
StepHypRef Expression
1 opnoncon.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2436 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 opnoncon.o . . . 4  |-  ._|_  =  ( oc `  K )
4 eqid 2436 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 opnoncon.m . . . 4  |-  ./\  =  ( meet `  K )
6 opnoncon.z . . . 4  |-  .0.  =  ( 0. `  K )
7 eqid 2436 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 29981 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X  ./\  (  ._|_  `  X )
)  =  .0.  )
)
983anidm23 1243 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X  ./\  (  ._|_  `  X )
)  =  .0.  )
)
109simp3d 971 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X ) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   occoc 13537   joincjn 14401   meetcmee 14402   0.cp0 14466   1.cp1 14467   OPcops 29970
This theorem is referenced by:  omlfh1N  30056  omlspjN  30059  atlatmstc  30117  pnonsingN  30730  lhpocnle  30813  dochnoncon  32189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-oposet 29974
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