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Theorem opposet 36311
Description: Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
Assertion
Ref Expression
opposet (𝐾 ∈ OP → 𝐾 ∈ Poset)

Proof of Theorem opposet
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2821 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2821 . . 3 (glb‘𝐾) = (glb‘𝐾)
4 eqid 2821 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2821 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2821 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2821 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2821 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2821 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 36310 . 2 (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ (Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((((oc‘𝐾)‘𝑥) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘((oc‘𝐾)‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝐾)𝑦 → ((oc‘𝐾)‘𝑦)(le‘𝐾)((oc‘𝐾)‘𝑥))) ∧ (𝑥(join‘𝐾)((oc‘𝐾)‘𝑥)) = (1.‘𝐾) ∧ (𝑥(meet‘𝐾)((oc‘𝐾)‘𝑥)) = (0.‘𝐾))))
11 simpl1 1187 . 2 (((𝐾 ∈ Poset ∧ (Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((((oc‘𝐾)‘𝑥) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘((oc‘𝐾)‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝐾)𝑦 → ((oc‘𝐾)‘𝑦)(le‘𝐾)((oc‘𝐾)‘𝑥))) ∧ (𝑥(join‘𝐾)((oc‘𝐾)‘𝑥)) = (1.‘𝐾) ∧ (𝑥(meet‘𝐾)((oc‘𝐾)‘𝑥)) = (0.‘𝐾))) → 𝐾 ∈ Poset)
1210, 11sylbi 219 1 (𝐾 ∈ OP → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138   class class class wbr 5058  dom cdm 5549  cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  occoc 16567  Posetcpo 17544  lubclub 17546  glbcglb 17547  joincjn 17548  meetcmee 17549  0.cp0 17641  1.cp1 17642  OPcops 36302
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-dm 5559  df-iota 6308  df-fv 6357  df-ov 7153  df-oposet 36306
This theorem is referenced by:  ople0  36317  op1le  36322  opltcon3b  36334  olposN  36345  ncvr1  36402  cvrcmp2  36414  leatb  36422  dalemcea  36790
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