Theorem op01dm 38655

Description: Conditions necessary for zero and unity elements to exist. (Contributed by NM, 14-Sep-2018.)

Hypotheses

Ref	Expression
op01dm.b	⊢ 𝐵 = (Base‘𝐾)
op01dm.u	⊢ 𝑈 = (lub‘𝐾)
op01dm.g	⊢ 𝐺 = (glb‘𝐾)

Assertion

Ref	Expression
op01dm	⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))

Proof of Theorem op01dm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		op01dm.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		op01dm.u	. . 3 ⊢ 𝑈 = (lub‘𝐾)
3		op01dm.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4		eqid 2728	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2728	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2728	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2728	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2728	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2728	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 38652	. 2 ⊢ (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((((oc‘𝐾)‘𝑥) ∈ 𝐵 ∧ ((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		simpl 482	. . 3 ⊢ (((𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((((oc‘𝐾)‘𝑥) ∈ 𝐵 ∧ ((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))) → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))
12	11	3adantl1 1164	. 2 ⊢ (((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((((oc‘𝐾)‘𝑥) ∈ 𝐵 ∧ ((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))) → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))
13	10, 12	sylbi 216	1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3058   class class class wbr 5148  dom cdm 5678  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  occoc 17241  Posetcpo 18299  lubclub 18301  glbcglb 18302  joincjn 18303  meetcmee 18304  0.cp0 18415  1.cp1 18416  OPcops 38644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-dm 5688  df-iota 6500  df-fv 6556  df-ov 7423  df-oposet 38648

This theorem is referenced by:  op0cl  38656  op1cl  38657  op0le  38658  ople1  38663  lhp2lt  39474