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Theorem opposet 33969
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 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2621 . . 3 (glb‘𝐾) = (glb‘𝐾)
4 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2621 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2621 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2621 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2621 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 33968 . 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 1062 . 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 207 1 (𝐾 ∈ OP → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4615  dom cdm 5076  cfv 5849  (class class class)co 6607  Basecbs 15784  lecple 15872  occoc 15873  Posetcpo 16864  lubclub 16866  glbcglb 16867  joincjn 16868  meetcmee 16869  0.cp0 16961  1.cp1 16962  OPcops 33960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-dm 5086  df-iota 5812  df-fv 5857  df-ov 6610  df-oposet 33964
This theorem is referenced by:  ople0  33975  op1le  33980  opltcon3b  33992  olposN  34003  ncvr1  34060  cvrcmp2  34072  leatb  34080  dalemcea  34447
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