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Theorem opoc1 34216
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 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2 opoc1.z . . . . . 6  |-  .0.  =  ( 0. `  K )
31, 2op0cl 34198 . . . . 5  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
4 opoc1.o . . . . . 6  |-  ._|_  =  ( oc `  K )
51, 4opoccl 34208 . . . . 5  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
) )  ->  (  ._|_  `  .0.  )  e.  ( Base `  K
) )
63, 5mpdan 668 . . . 4  |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  e.  ( Base `  K
) )
7 eqid 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
8 opoc1.u . . . . 5  |-  .1.  =  ( 1. `  K )
91, 7, 8ople1 34205 . . . 4  |-  ( ( K  e.  OP  /\  (  ._|_  `  .0.  )  e.  ( Base `  K
) )  ->  (  ._|_  `  .0.  ) ( le `  K )  .1.  )
106, 9mpdan 668 . . 3  |-  ( K  e.  OP  ->  (  ._|_  `  .0.  ) ( le `  K )  .1.  )
111, 8op1cl 34199 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
121, 7, 4oplecon1b 34215 . . . 4  |-  ( ( K  e.  OP  /\  .1.  e.  ( Base `  K
)  /\  .0.  e.  ( Base `  K )
)  ->  ( (  ._|_  `  .1.  ) ( le `  K )  .0.  <->  (  ._|_  `  .0.  ) ( le `  K )  .1.  )
)
1311, 3, 12mpd3an23 1326 . . 3  |-  ( K  e.  OP  ->  (
(  ._|_  `  .1.  )
( le `  K
)  .0.  <->  (  ._|_  `  .0.  ) ( le
`  K )  .1.  ) )
1410, 13mpbird 232 . 2  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  ) ( le `  K )  .0.  )
151, 4opoccl 34208 . . . 4  |-  ( ( K  e.  OP  /\  .1.  e.  ( Base `  K
) )  ->  (  ._|_  `  .1.  )  e.  ( Base `  K
) )
1611, 15mpdan 668 . . 3  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  e.  ( Base `  K
) )
171, 7, 2ople0 34201 . . 3  |-  ( ( K  e.  OP  /\  (  ._|_  `  .1.  )  e.  ( Base `  K
) )  ->  (
(  ._|_  `  .1.  )
( le `  K
)  .0.  <->  (  ._|_  `  .1.  )  =  .0.  ) )
1816, 17mpdan 668 . 2  |-  ( K  e.  OP  ->  (
(  ._|_  `  .1.  )
( le `  K
)  .0.  <->  (  ._|_  `  .1.  )  =  .0.  ) )
1914, 18mpbid 210 1  |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588   Basecbs 14493   lecple 14565   occoc 14566   0.cp0 15527   1.cp1 15528   OPcops 34186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-poset 15436  df-lub 15464  df-glb 15465  df-p0 15529  df-p1 15530  df-oposet 34190
This theorem is referenced by:  opoc0  34217  olm11  34241  1cvrco  34485  1cvrjat  34488  pol1N  34923  doch1  36373
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