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Theorem opococ 29385
Description: Double negative law for orthoposets. (ococ 21985 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 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 opoccl.o . . . . 5  |-  ._|_  =  ( oc `  K )
4 eqid 2283 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2283 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2283 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2283 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 29373 . . . 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 1241 . . 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 967 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  ._|_  `  X
)  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K
) X  ->  (  ._|_  `  X ) ( le `  K ) (  ._|_  `  X ) ) ) )
1110simp2d 968 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   0.cp0 14143   1.cp1 14144   OPcops 29362
This theorem is referenced by:  opcon3b  29386  opcon2b  29387  oplecon3b  29390  oplecon1b  29391  opltcon1b  29395  opltcon2b  29396  oldmm2  29408  oldmm3N  29409  oldmm4  29410  oldmj1  29411  oldmj2  29412  oldmj3  29413  oldmj4  29414  olm11  29417  omllaw4  29436  cmt2N  29440  glbconN  29566  1cvratex  29662  1cvrjat  29664  polval2N  30095  2polpmapN  30102  2polvalN  30103  2polatN  30121  lhpoc2N  30204  doch2val2  31554  dochocss  31556  dochoc  31557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-oposet 29366
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