Theorem op01dm 39382

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 2734	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2734	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6		eqid 2734	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2734	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		eqid 2734	. . 3 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		eqid 2734	. . 3 ⊢ (1.‘𝐾) = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 39379	. 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 1167	. 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 217	1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   class class class wbr 5096  dom cdm 5622  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  lecple 17182  occoc 17183  Posetcpo 18228  lubclub 18230  glbcglb 18231  joincjn 18232  meetcmee 18233  0.cp0 18342  1.cp1 18343  OPcops 39371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-dm 5632  df-iota 6446  df-fv 6498  df-ov 7359  df-oposet 39375

This theorem is referenced by:  op0cl  39383  op1cl  39384  op0le  39385  ople1  39390  lhp2lt  40200