Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opococ Structured version   Visualization version   GIF version

Theorem opococ 38065
Description: Double negative law for orthoposets. (ococ 30659 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opoccl.b 𝐡 = (Baseβ€˜πΎ)
opoccl.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
opococ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)

Proof of Theorem opococ
StepHypRef Expression
1 opoccl.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 eqid 2733 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
3 opoccl.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
4 eqid 2733 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
5 eqid 2733 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
6 eqid 2733 . . . . 5 (0.β€˜πΎ) = (0.β€˜πΎ)
7 eqid 2733 . . . . 5 (1.β€˜πΎ) = (1.β€˜πΎ)
81, 2, 3, 4, 5, 6, 7oposlem 38052 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋(leβ€˜πΎ)𝑋 β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹))) ∧ (𝑋(joinβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (1.β€˜πΎ) ∧ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (0.β€˜πΎ)))
983anidm23 1422 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋(leβ€˜πΎ)𝑋 β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹))) ∧ (𝑋(joinβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (1.β€˜πΎ) ∧ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (0.β€˜πΎ)))
109simp1d 1143 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋(leβ€˜πΎ)𝑋 β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹))))
1110simp2d 1144 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  occoc 17205  joincjn 18264  meetcmee 18265  0.cp0 18376  1.cp1 18377  OPcops 38042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412  df-oposet 38046
This theorem is referenced by:  opcon3b  38066  opcon2b  38067  oplecon3b  38070  oplecon1b  38071  opltcon1b  38075  opltcon2b  38076  oldmm2  38088  oldmm3N  38089  oldmm4  38090  oldmj1  38091  oldmj2  38092  oldmj3  38093  oldmj4  38094  olm11  38097  omllaw4  38116  cmt2N  38120  glbconN  38247  glbconNOLD  38248  1cvratex  38344  1cvrjat  38346  polval2N  38777  2polpmapN  38784  2polvalN  38785  2polatN  38803  lhpoc2N  38886  doch2val2  40235  dochocss  40237  dochoc  40238
  Copyright terms: Public domain W3C validator