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 34990
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 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2771 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2771 . . 3 (glb‘𝐾) = (glb‘𝐾)
4 eqid 2771 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2771 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2771 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2771 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2771 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2771 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 34989 . 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 1227 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4787  dom cdm 5250  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  occoc 16157  Posetcpo 17148  lubclub 17150  glbcglb 17151  joincjn 17152  meetcmee 17153  0.cp0 17245  1.cp1 17246  OPcops 34981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-dm 5260  df-iota 5993  df-fv 6038  df-ov 6799  df-oposet 34985
This theorem is referenced by:  ople0  34996  op1le  35001  opltcon3b  35013  olposN  35024  ncvr1  35081  cvrcmp2  35093  leatb  35101  dalemcea  35469
  Copyright terms: Public domain W3C validator