Theorem opposet 39769

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.

Step	Hyp	Ref	Expression
1		eqid 2761	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2761	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2761	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		eqid 2761	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2761	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2761	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2761	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2761	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2761	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 39768	. 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 1204	. 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)
12	10, 11	sylbi 219	1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5099  dom cdm 5645  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  lecple 17276  occoc 17277  Posetcpo 18322  lubclub 18324  glbcglb 18325  joincjn 18326  meetcmee 18327  0.cp0 18436  1.cp1 18437  OPcops 39760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-dm 5655  df-iota 6473  df-fv 6525  df-ov 7395  df-oposet 39764

This theorem is referenced by:  ople0  39775  op1le  39780  opltcon3b  39792  olposN  39803  ncvr1  39860  cvrcmp2  39872  leatb  39880  dalemcea  40248