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Theorem opococ 33989
Description: Double negative law for orthoposets. (ococ 28132 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opococ ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)

Proof of Theorem opococ
StepHypRef Expression
1 opoccl.b . . . . 5 𝐵 = (Base‘𝐾)
2 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 opoccl.o . . . . 5 = (oc‘𝐾)
4 eqid 2621 . . . . 5 (join‘𝐾) = (join‘𝐾)
5 eqid 2621 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2621 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2621 . . . . 5 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 33976 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑋𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))) ∧ (𝑋(join‘𝐾)( 𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( 𝑋)) = (0.‘𝐾)))
983anidm23 1382 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))) ∧ (𝑋(join‘𝐾)( 𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( 𝑋)) = (0.‘𝐾)))
109simp1d 1071 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))))
1110simp2d 1072 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15788  lecple 15876  occoc 15877  joincjn 16872  meetcmee 16873  0.cp0 16965  1.cp1 16966  OPcops 33966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-dm 5089  df-iota 5815  df-fv 5860  df-ov 6613  df-oposet 33970
This theorem is referenced by:  opcon3b  33990  opcon2b  33991  oplecon3b  33994  oplecon1b  33995  opltcon1b  33999  opltcon2b  34000  oldmm2  34012  oldmm3N  34013  oldmm4  34014  oldmj1  34015  oldmj2  34016  oldmj3  34017  oldmj4  34018  olm11  34021  omllaw4  34040  cmt2N  34044  glbconN  34170  1cvratex  34266  1cvrjat  34268  polval2N  34699  2polpmapN  34706  2polvalN  34707  2polatN  34725  lhpoc2N  34808  doch2val2  36160  dochocss  36162  dochoc  36163
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