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Theorem opococ 39196
Description: Double negative law for orthoposets. (ococ 31425 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 2737 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 opoccl.o . . . . 5 = (oc‘𝐾)
4 eqid 2737 . . . . 5 (join‘𝐾) = (join‘𝐾)
5 eqid 2737 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2737 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2737 . . . . 5 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 39183 . . . 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 1423 . . 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 1143 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))))
1110simp2d 1144 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  0.cp0 18468  1.cp1 18469  OPcops 39173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434  df-oposet 39177
This theorem is referenced by:  opcon3b  39197  opcon2b  39198  oplecon3b  39201  oplecon1b  39202  opltcon1b  39206  opltcon2b  39207  oldmm2  39219  oldmm3N  39220  oldmm4  39221  oldmj1  39222  oldmj2  39223  oldmj3  39224  oldmj4  39225  olm11  39228  omllaw4  39247  cmt2N  39251  glbconN  39378  glbconNOLD  39379  1cvratex  39475  1cvrjat  39477  polval2N  39908  2polpmapN  39915  2polvalN  39916  2polatN  39934  lhpoc2N  40017  doch2val2  41366  dochocss  41368  dochoc  41369
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