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Theorem opococ 34994
Description: Double negative law for orthoposets. (ococ 28616 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 2817 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 opoccl.o . . . . 5 = (oc‘𝐾)
4 eqid 2817 . . . . 5 (join‘𝐾) = (join‘𝐾)
5 eqid 2817 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2817 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2817 . . . . 5 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 34981 . . . 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 1537 . . 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 1165 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))))
1110simp2d 1166 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157   class class class wbr 4855  cfv 6111  (class class class)co 6884  Basecbs 16088  lecple 16180  occoc 16181  joincjn 17169  meetcmee 17170  0.cp0 17262  1.cp1 17263  OPcops 34971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-nul 4996
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-dm 5334  df-iota 6074  df-fv 6119  df-ov 6887  df-oposet 34975
This theorem is referenced by:  opcon3b  34995  opcon2b  34996  oplecon3b  34999  oplecon1b  35000  opltcon1b  35004  opltcon2b  35005  oldmm2  35017  oldmm3N  35018  oldmm4  35019  oldmj1  35020  oldmj2  35021  oldmj3  35022  oldmj4  35023  olm11  35026  omllaw4  35045  cmt2N  35049  glbconN  35176  1cvratex  35272  1cvrjat  35274  polval2N  35705  2polpmapN  35712  2polvalN  35713  2polatN  35731  lhpoc2N  35814  doch2val2  37163  dochocss  37165  dochoc  37166
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