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

Theorem opposet 39183
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 2736 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2736 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2736 . . 3 (glb‘𝐾) = (glb‘𝐾)
4 eqid 2736 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2736 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2736 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2736 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2736 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2736 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 39182 . 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 1191 . 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 217 1 (𝐾 ∈ OP → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142  dom cdm 5684  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  occoc 17306  Posetcpo 18354  lubclub 18356  glbcglb 18357  joincjn 18358  meetcmee 18359  0.cp0 18469  1.cp1 18470  OPcops 39174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-dm 5694  df-iota 6513  df-fv 6568  df-ov 7435  df-oposet 39178
This theorem is referenced by:  ople0  39189  op1le  39194  opltcon3b  39206  olposN  39217  ncvr1  39274  cvrcmp2  39286  leatb  39294  dalemcea  39663
  Copyright terms: Public domain W3C validator