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Theorem opoc1 29318
Description: Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
Hypotheses
Ref Expression
opoc1.z  |-  .0.  =  ( 0. `  K )
opoc1.u  |-  .1.  =  ( 1. `  K )
opoc1.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
opoc1  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )

Proof of Theorem opoc1
StepHypRef Expression
1 eqid 2388 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2 opoc1.z . . . . . 6  |-  .0.  =  ( 0. `  K )
31, 2op0cl 29300 . . . . 5  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
4 opoc1.o . . . . . 6  |-  ._|_  =  ( oc `  K )
51, 4opoccl 29310 . . . . 5  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
) )  ->  (  ._|_  `  .0.  )  e.  ( Base `  K
) )
63, 5mpdan 650 . . . 4  |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  e.  ( Base `  K
) )
7 eqid 2388 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
8 opoc1.u . . . . 5  |-  .1.  =  ( 1. `  K )
91, 7, 8ople1 29307 . . . 4  |-  ( ( K  e.  OP  /\  (  ._|_  `  .0.  )  e.  ( Base `  K
) )  ->  (  ._|_  `  .0.  ) ( le `  K )  .1.  )
106, 9mpdan 650 . . 3  |-  ( K  e.  OP  ->  (  ._|_  `  .0.  ) ( le `  K )  .1.  )
111, 8op1cl 29301 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
121, 7, 4oplecon1b 29317 . . . 4  |-  ( ( K  e.  OP  /\  .1.  e.  ( Base `  K
)  /\  .0.  e.  ( Base `  K )
)  ->  ( (  ._|_  `  .1.  ) ( le `  K )  .0.  <->  (  ._|_  `  .0.  ) ( le `  K )  .1.  )
)
1311, 3, 12mpd3an23 1281 . . 3  |-  ( K  e.  OP  ->  (
(  ._|_  `  .1.  )
( le `  K
)  .0.  <->  (  ._|_  `  .0.  ) ( le
`  K )  .1.  ) )
1410, 13mpbird 224 . 2  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  ) ( le `  K )  .0.  )
151, 4opoccl 29310 . . . 4  |-  ( ( K  e.  OP  /\  .1.  e.  ( Base `  K
) )  ->  (  ._|_  `  .1.  )  e.  ( Base `  K
) )
1611, 15mpdan 650 . . 3  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  e.  ( Base `  K
) )
171, 7, 2ople0 29303 . . 3  |-  ( ( K  e.  OP  /\  (  ._|_  `  .1.  )  e.  ( Base `  K
) )  ->  (
(  ._|_  `  .1.  )
( le `  K
)  .0.  <->  (  ._|_  `  .1.  )  =  .0.  ) )
1816, 17mpdan 650 . 2  |-  ( K  e.  OP  ->  (
(  ._|_  `  .1.  )
( le `  K
)  .0.  <->  (  ._|_  `  .1.  )  =  .0.  ) )
1914, 18mpbid 202 1  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   occoc 13465   0.cp0 14394   1.cp1 14395   OPcops 29288
This theorem is referenced by:  opoc0  29319  olm11  29343  1cvrco  29587  1cvrjat  29590  pol1N  30025  doch1  31475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-undef 6480  df-riota 6486  df-poset 14331  df-lub 14359  df-glb 14360  df-p0 14396  df-p1 14397  df-oposet 29292
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