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Theorem opexmid 29942
Description: Law of excluded middle for orthoposets. (chjo 23009 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 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 opexmid.o . . . 4  |-  ._|_  =  ( oc `  K )
4 opexmid.j . . . 4  |-  .\/  =  ( join `  K )
5 eqid 2435 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2435 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 opexmid.u . . . 4  |-  .1.  =  ( 1. `  K )
81, 2, 3, 4, 5, 6, 7oposlem 29918 . . 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 1243 . 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 970 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .\/  (  ._|_  `  X ) )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   occoc 13529   joincjn 14393   meetcmee 14394   0.cp0 14458   1.cp1 14459   OPcops 29907
This theorem is referenced by:  dih1  32021
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 2416  ax-nul 4330
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 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-oposet 29911
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