Theorem op01dm 39819

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 2765	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2765	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2765	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2765	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2765	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2765	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 39816	. 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 487	. . 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 1183	. 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 220	1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079   class class class wbr 5105  dom cdm 5652  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  lecple 17307  occoc 17308  Posetcpo 18353  lubclub 18355  glbcglb 18356  joincjn 18357  meetcmee 18358  0.cp0 18467  1.cp1 18468  OPcops 39808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-dm 5662  df-iota 6481  df-fv 6533  df-ov 7403  df-oposet 39812

This theorem is referenced by:  op0cl  39820  op1cl  39821  op0le  39822  ople1  39827  lhp2lt  40637