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Theorem opococ 30007
Description: Double negative law for orthoposets. (ococ 22001 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 2296 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 opoccl.o . . . . 5  |-  ._|_  =  ( oc `  K )
4 eqid 2296 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2296 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2296 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2296 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 29995 . . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   meetcmee 14095   0.cp0 14159   1.cp1 14160   OPcops 29984
This theorem is referenced by:  opcon3b  30008  opcon2b  30009  oplecon3b  30012  oplecon1b  30013  opltcon1b  30017  opltcon2b  30018  oldmm2  30030  oldmm3N  30031  oldmm4  30032  oldmj1  30033  oldmj2  30034  oldmj3  30035  oldmj4  30036  olm11  30039  omllaw4  30058  cmt2N  30062  glbconN  30188  1cvratex  30284  1cvrjat  30286  polval2N  30717  2polpmapN  30724  2polvalN  30725  2polatN  30743  lhpoc2N  30826  doch2val2  32176  dochocss  32178  dochoc  32179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-oposet 29988
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