Theorem opposet 40062

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 2762	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2762	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2762	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		eqid 2762	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2762	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2762	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2762	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2762	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2762	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 40061	. 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 1210	. 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 220	1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  lecple 17355  occoc 17356  Posetcpo 18401  lubclub 18403  glbcglb 18404  joincjn 18405  meetcmee 18406  0.cp0 18515  1.cp1 18516  OPcops 40053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7420  df-oposet 40057

This theorem is used by:  ople0  40068  op1le  40073  opltcon3b  40085  olposN  40096  ncvr1  40153  cvrcmp2  40165  leatb  40173  dalemcea  40541