Theorem opposet 39180

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 2729	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2729	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2729	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		eqid 2729	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2729	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2729	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2729	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2729	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2729	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 39179	. 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 1192	. 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 217	1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5092  dom cdm 5619  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  occoc 17169  Posetcpo 18213  lubclub 18215  glbcglb 18216  joincjn 18217  meetcmee 18218  0.cp0 18327  1.cp1 18328  OPcops 39171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5245

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-dm 5629  df-iota 6438  df-fv 6490  df-ov 7352  df-oposet 39175

This theorem is referenced by:  ople0  39186  op1le  39191  opltcon3b  39203  olposN  39214  ncvr1  39271  cvrcmp2  39283  leatb  39291  dalemcea  39659