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Theorem opococ 34209
Description: Double negative law for orthoposets. (ococ 26097 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opoccl.b  |-  B  =  ( Base `  K
)
opoccl.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
opococ  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )

Proof of Theorem opococ
StepHypRef Expression
1 opoccl.b . . . . 5  |-  B  =  ( Base `  K
)
2 eqid 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 opoccl.o . . . . 5  |-  ._|_  =  ( oc `  K )
4 eqid 2467 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2467 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2467 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2467 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 34196 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X (
meet `  K )
(  ._|_  `  X )
)  =  ( 0.
`  K ) ) )
983anidm23 1287 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X (
meet `  K )
(  ._|_  `  X )
)  =  ( 0.
`  K ) ) )
109simp1d 1008 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  ._|_  `  X
)  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K
) X  ->  (  ._|_  `  X ) ( le `  K ) (  ._|_  `  X ) ) ) )
1110simp2d 1009 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   occoc 14566   joincjn 15434   meetcmee 15435   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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009  df-iota 5551  df-fv 5596  df-ov 6288  df-oposet 34190
This theorem is referenced by:  opcon3b  34210  opcon2b  34211  oplecon3b  34214  oplecon1b  34215  opltcon1b  34219  opltcon2b  34220  oldmm2  34232  oldmm3N  34233  oldmm4  34234  oldmj1  34235  oldmj2  34236  oldmj3  34237  oldmj4  34238  olm11  34241  omllaw4  34260  cmt2N  34264  glbconN  34390  1cvratex  34486  1cvrjat  34488  polval2N  34919  2polpmapN  34926  2polvalN  34927  2polatN  34945  lhpoc2N  35028  doch2val2  36378  dochocss  36380  dochoc  36381
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