Theorem isopos 39218

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 6822	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2		isopos.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2784	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4		fveq2 6822	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5		isopos.e	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝐾)
6	4, 5	eqtr4di 2784	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
7	6	dmeqd 5845	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
8	3, 7	eleq12d 2825	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9		fveq2 6822	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10		isopos.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
11	9, 10	eqtr4di 2784	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
12	11	dmeqd 5845	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
13	3, 12	eleq12d 2825	. . . . 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 6822	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16		isopos.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
17	15, 16	eqtr4di 2784	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = ⊥ )
18	17	eqeq2d 2742	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ⊥ ))
19	3	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → ((𝑛‘𝑥) ∈ (Base‘𝑝) ↔ (𝑛‘𝑥) ∈ 𝐵))
20		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21		isopos.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
22	20, 21	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
23	22	breqd 5102	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦 ↔ 𝑥 ≤ 𝑦))
24	22	breqd 5102	. . . . . . . . . . 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 6822	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28		isopos.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
29	27, 28	eqtr4di 2784	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = ∨ )
30	29	oveqd 7363	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛‘𝑥)) = (𝑥 ∨ (𝑛‘𝑥)))
31		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32		isopos.u	. . . . . . . . . . 11 ⊢ 1 = (1.‘𝐾)
33	31, 32	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
34	30, 33	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ↔ (𝑥 ∨ (𝑛‘𝑥)) = 1 ))
35		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36		isopos.m	. . . . . . . . . . . 12 ⊢ ∧ = (meet‘𝐾)
37	35, 36	eqtr4di 2784	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = ∧ )
38	37	oveqd 7363	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (𝑥 ∧ (𝑛‘𝑥)))
39		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40		isopos.f	. . . . . . . . . . 11 ⊢ 0 = (0.‘𝐾)
41	39, 40	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
42	38, 41	eqeq12d 2747	. . . . . . . . 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 3312	. . . . . . 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 3312	. . . . . 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 39214	. . 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 3650	. 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 6836	. . . 4 ⊢ ⊥ ∈ V
55		fveq1 6821	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘𝑥) = ( ⊥ ‘𝑥))
56	55	eleq1d 2816	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑥) ∈ 𝐵))
57		id 22	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → 𝑛 = ⊥ )
58	57, 55	fveq12d 6829	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘(𝑛‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑥)))
59	58	eqeq1d 2733	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘(𝑛‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
60		fveq1 6821	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → (𝑛‘𝑦) = ( ⊥ ‘𝑦))
61	60, 55	breq12d 5104	. . . . . . . 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 7362	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∨ (𝑛‘𝑥)) = (𝑥 ∨ ( ⊥ ‘𝑥)))
65	64	eqeq1d 2733	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∨ (𝑛‘𝑥)) = 1 ↔ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ))
66	55	oveq2d 7362	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∧ (𝑛‘𝑥)) = (𝑥 ∧ ( ⊥ ‘𝑥)))
67	66	eqeq1d 2733	. . . . . 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 3196	. . . 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 3487	. . 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 2111  ∀wral 3047   class class class wbr 5091  dom cdm 5616  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  lecple 17165  occoc 17166  Posetcpo 18210  lubclub 18212  glbcglb 18213  joincjn 18214  meetcmee 18215  0.cp0 18324  1.cp1 18325  OPcops 39210

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-dm 5626  df-iota 6437  df-fv 6489  df-ov 7349  df-oposet 39214

This theorem is referenced by:  opposet  39219  oposlem  39220  op01dm  39221