Theorem isopos 39436

Description: The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)

Hypotheses

Ref	Expression
isopos.b	⊢ 𝐵 = (Base‘𝐾)
isopos.e	⊢ 𝑈 = (lub‘𝐾)
isopos.g	⊢ 𝐺 = (glb‘𝐾)
isopos.l	⊢ ≤ = (le‘𝐾)
isopos.o	⊢ ⊥ = (oc‘𝐾)
isopos.j	⊢ ∨ = (join‘𝐾)
isopos.m	⊢ ∧ = (meet‘𝐾)
isopos.f	⊢ 0 = (0.‘𝐾)
isopos.u	⊢ 1 = (1.‘𝐾)

Assertion

Ref	Expression
isopos	⊢ (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥, ⊥ ,𝑦   𝑥,𝐾,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   1 (𝑥,𝑦)   𝐺(𝑥,𝑦)   ∨ (𝑥,𝑦)   ≤ (𝑥,𝑦)   ∧ (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isopos

Dummy variables 𝑛 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2		isopos.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2789	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5		isopos.e	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝐾)
6	4, 5	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
7	6	dmeqd 5854	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
8	3, 7	eleq12d 2830	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10		isopos.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
11	9, 10	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
12	11	dmeqd 5854	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
13	3, 12	eleq12d 2830	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
14	8, 13	anbi12d 632	. . . 4 ⊢ (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)))
15		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16		isopos.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
17	15, 16	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = ⊥ )
18	17	eqeq2d 2747	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ⊥ ))
19	3	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → ((𝑛‘𝑥) ∈ (Base‘𝑝) ↔ (𝑛‘𝑥) ∈ 𝐵))
20		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21		isopos.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
22	20, 21	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
23	22	breqd 5109	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦 ↔ 𝑥 ≤ 𝑦))
24	22	breqd 5109	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → ((𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥) ↔ (𝑛‘𝑦) ≤ (𝑛‘𝑥)))
25	23, 24	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → ((𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥)) ↔ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))))
26	19, 25	3anbi13d 1440	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ↔ ((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥)))))
27		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28		isopos.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
29	27, 28	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = ∨ )
30	29	oveqd 7375	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛‘𝑥)) = (𝑥 ∨ (𝑛‘𝑥)))
31		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32		isopos.u	. . . . . . . . . . 11 ⊢ 1 = (1.‘𝐾)
33	31, 32	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
34	30, 33	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ↔ (𝑥 ∨ (𝑛‘𝑥)) = 1 ))
35		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36		isopos.m	. . . . . . . . . . . 12 ⊢ ∧ = (meet‘𝐾)
37	35, 36	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = ∧ )
38	37	oveqd 7375	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (𝑥 ∧ (𝑛‘𝑥)))
39		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40		isopos.f	. . . . . . . . . . 11 ⊢ 0 = (0.‘𝐾)
41	39, 40	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
42	38, 41	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝) ↔ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))
43	26, 34, 42	3anbi123d 1438	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ((((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝)) ↔ (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )))
44	3, 43	raleqbidv 3316	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∀𝑦 ∈ (Base‘𝑝)(((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝)) ↔ ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )))
45	3, 44	raleqbidv 3316	. . . . . 6 ⊢ (𝑝 = 𝐾 → (∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )))
46	18, 45	anbi12d 632	. . . . 5 ⊢ (𝑝 = 𝐾 → ((𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝))) ↔ (𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))))
47	46	exbidv 1922	. . . 4 ⊢ (𝑝 = 𝐾 → (∃𝑛(𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝))) ↔ ∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))))
48	14, 47	anbi12d 632	. . 3 ⊢ (𝑝 = 𝐾 → ((((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑛(𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝)))) ↔ ((𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )))))
49		df-oposet 39432	. . 3 ⊢ OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑛(𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛‘𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥))) ∧ (𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝))))}
50	48, 49	elrab2 3649	. 2 ⊢ (𝐾 ∈ OP ↔ (𝐾 ∈ Poset ∧ ((𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )))))
51		anass 468	. 2 ⊢ (((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) ∧ ∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))) ↔ (𝐾 ∈ Poset ∧ ((𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )))))
52		3anass 1094	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)))
53	52	bicomi 224	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) ↔ (𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))
54	16	fvexi 6848	. . . 4 ⊢ ⊥ ∈ V
55		fveq1 6833	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘𝑥) = ( ⊥ ‘𝑥))
56	55	eleq1d 2821	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑥) ∈ 𝐵))
57		id 22	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → 𝑛 = ⊥ )
58	57, 55	fveq12d 6841	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘(𝑛‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑥)))
59	58	eqeq1d 2738	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘(𝑛‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
60		fveq1 6833	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → (𝑛‘𝑦) = ( ⊥ ‘𝑦))
61	60, 55	breq12d 5111	. . . . . . . 8 ⊢ (𝑛 = ⊥ → ((𝑛‘𝑦) ≤ (𝑛‘𝑥) ↔ ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥)))
62	61	imbi2d 340	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥)) ↔ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))))
63	56, 59, 62	3anbi123d 1438	. . . . . 6 ⊢ (𝑛 = ⊥ → (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ↔ (( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥)))))
64	55	oveq2d 7374	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∨ (𝑛‘𝑥)) = (𝑥 ∨ ( ⊥ ‘𝑥)))
65	64	eqeq1d 2738	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∨ (𝑛‘𝑥)) = 1 ↔ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ))
66	55	oveq2d 7374	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∧ (𝑛‘𝑥)) = (𝑥 ∧ ( ⊥ ‘𝑥)))
67	66	eqeq1d 2738	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∧ (𝑛‘𝑥)) = 0 ↔ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ))
68	63, 65, 67	3anbi123d 1438	. . . . 5 ⊢ (𝑛 = ⊥ → ((((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 ) ↔ ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))
69	68	2ralbidv 3200	. . . 4 ⊢ (𝑛 = ⊥ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))
70	54, 69	ceqsexv 3490	. . 3 ⊢ (∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 )) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ))
71	53, 70	anbi12i 628	. 2 ⊢ (((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) ∧ ∃𝑛(𝑛 = ⊥ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑛‘𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))) ∧ (𝑥 ∨ (𝑛‘𝑥)) = 1 ∧ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))) ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))
72	50, 51, 71	3bitr2i 299	1 ⊢ (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  occoc 17185  Posetcpo 18230  lubclub 18232  glbcglb 18233  joincjn 18234  meetcmee 18235  0.cp0 18344  1.cp1 18345  OPcops 39428

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 2708  ax-nul 5251

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-dm 5634  df-iota 6448  df-fv 6500  df-ov 7361  df-oposet 39432

This theorem is referenced by:  opposet  39437  oposlem  39438  op01dm  39439