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Theorem opexmid 35329
Description: Law of excluded middle for orthoposets. (chjo 26631 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 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 opexmid.o . . . 4  |-  ._|_  =  ( oc `  K )
4 opexmid.j . . . 4  |-  .\/  =  ( join `  K )
5 eqid 2454 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2454 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 opexmid.u . . . 4  |-  .1.  =  ( 1. `  K )
81, 2, 3, 4, 5, 6, 7oposlem 35304 . . 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 1285 . 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 1007 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .\/  (  ._|_  `  X ) )  =  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   occoc 14792   joincjn 15772   meetcmee 15773   0.cp0 15866   1.cp1 15867   OPcops 35294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-dm 4998  df-iota 5534  df-fv 5578  df-ov 6273  df-oposet 35298
This theorem is referenced by:  dih1  37410
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