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Theorem opococ 39177
Description: Double negative law for orthoposets. (ococ 31435 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 2735 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 opoccl.o . . . . 5 = (oc‘𝐾)
4 eqid 2735 . . . . 5 (join‘𝐾) = (join‘𝐾)
5 eqid 2735 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2735 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2735 . . . . 5 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 39164 . . . 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 1420 . . 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 1141 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))))
1110simp2d 1142 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  0.cp0 18481  1.cp1 18482  OPcops 39154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oposet 39158
This theorem is referenced by:  opcon3b  39178  opcon2b  39179  oplecon3b  39182  oplecon1b  39183  opltcon1b  39187  opltcon2b  39188  oldmm2  39200  oldmm3N  39201  oldmm4  39202  oldmj1  39203  oldmj2  39204  oldmj3  39205  oldmj4  39206  olm11  39209  omllaw4  39228  cmt2N  39232  glbconN  39359  glbconNOLD  39360  1cvratex  39456  1cvrjat  39458  polval2N  39889  2polpmapN  39896  2polvalN  39897  2polatN  39915  lhpoc2N  39998  doch2val2  41347  dochocss  41349  dochoc  41350
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