Theorem opposet 39680

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 2740	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2740	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2740	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		eqid 2740	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2740	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2740	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2740	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2740	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2740	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 39679	. 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 1198	. 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 218	1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054   class class class wbr 5079  dom cdm 5625  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  lecple 17225  occoc 17226  Posetcpo 18271  lubclub 18273  glbcglb 18274  joincjn 18275  meetcmee 18276  0.cp0 18385  1.cp1 18386  OPcops 39671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-dm 5635  df-iota 6448  df-fv 6500  df-ov 7366  df-oposet 39675

This theorem is referenced by:  ople0  39686  op1le  39691  opltcon3b  39703  olposN  39714  ncvr1  39771  cvrcmp2  39783  leatb  39791  dalemcea  40159