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Theorem opococ 38370
Description: Double negative law for orthoposets. (ococ 30924 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 2730 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
3 opoccl.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
4 eqid 2730 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
5 eqid 2730 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
6 eqid 2730 . . . . 5 (0.β€˜πΎ) = (0.β€˜πΎ)
7 eqid 2730 . . . . 5 (1.β€˜πΎ) = (1.β€˜πΎ)
81, 2, 3, 4, 5, 6, 7oposlem 38357 . . . 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 1419 . . 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 1140 . 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 1141 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  Basecbs 17150  lecple 17210  occoc 17211  joincjn 18270  meetcmee 18271  0.cp0 18382  1.cp1 18383  OPcops 38347
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  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 7416  df-oposet 38351
This theorem is referenced by:  opcon3b  38371  opcon2b  38372  oplecon3b  38375  oplecon1b  38376  opltcon1b  38380  opltcon2b  38381  oldmm2  38393  oldmm3N  38394  oldmm4  38395  oldmj1  38396  oldmj2  38397  oldmj3  38398  oldmj4  38399  olm11  38402  omllaw4  38421  cmt2N  38425  glbconN  38552  glbconNOLD  38553  1cvratex  38649  1cvrjat  38651  polval2N  39082  2polpmapN  39089  2polvalN  39090  2polatN  39108  lhpoc2N  39191  doch2val2  40540  dochocss  40542  dochoc  40543
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