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Theorem opexmid 28770
Description: Law of excluded middle for orthoposets. (chjo 22094 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opexmid.b  |-  B  =  ( Base `  K
)
opexmid.o  |-  ._|_  =  ( oc `  K )
opexmid.j  |-  .\/  =  ( join `  K )
opexmid.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
opexmid  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .\/  (  ._|_  `  X ) )  =  .1.  )

Proof of Theorem opexmid
StepHypRef Expression
1 opexmid.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 opexmid.o . . . 4  |-  ._|_  =  ( oc `  K )
4 opexmid.j . . . 4  |-  .\/  =  ( join `  K )
5 eqid 2283 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2283 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 opexmid.u . . . 4  |-  .1.  =  ( 1. `  K )
81, 2, 3, 4, 5, 6, 7oposlem 28746 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X  .\/  (  ._|_  `  X ) )  =  .1.  /\  ( X ( meet `  K
) (  ._|_  `  X
) )  =  ( 0. `  K ) ) )
983anidm23 1241 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X  .\/  (  ._|_  `  X ) )  =  .1.  /\  ( X ( meet `  K
) (  ._|_  `  X
) )  =  ( 0. `  K ) ) )
109simp2d 968 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .\/  (  ._|_  `  X ) )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   0.cp0 14143   1.cp1 14144   OPcops 28735
This theorem is referenced by:  dih1  30849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-oposet 28739
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