Theorem isopos 39640

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 2790	. . . . . 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 2790	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
7	6	dmeqd 5854	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
8	3, 7	eleq12d 2831	. . . . 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 2790	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
12	11	dmeqd 5854	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
13	3, 12	eleq12d 2831	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
14	8, 13	anbi12d 633	. . . 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 2790	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = ⊥ )
18	17	eqeq2d 2748	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ⊥ ))
19	3	eleq2d 2823	. . . . . . . . . 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 2790	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
23	22	breqd 5097	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦 ↔ 𝑥 ≤ 𝑦))
24	22	breqd 5097	. . . . . . . . . . 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 1441	. . . . . . . . 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 2790	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = ∨ )
30	29	oveqd 7377	. . . . . . . . . 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 2790	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
34	30, 33	eqeq12d 2753	. . . . . . . . 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 2790	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = ∧ )
38	37	oveqd 7377	. . . . . . . . . 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 2790	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
42	38, 41	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝) ↔ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))
43	26, 34, 42	3anbi123d 1439	. . . . . . . 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 633	. . . . 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 1923	. . . 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 633	. . 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 39636	. . 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 3638	. 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 1095	. . . 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 2822	. . . . . . 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 2739	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘(𝑛‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
60		fveq1 6833	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → (𝑛‘𝑦) = ( ⊥ ‘𝑦))
61	60, 55	breq12d 5099	. . . . . . . 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 1439	. . . . . 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 7376	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∨ (𝑛‘𝑥)) = (𝑥 ∨ ( ⊥ ‘𝑥)))
65	64	eqeq1d 2739	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∨ (𝑛‘𝑥)) = 1 ↔ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ))
66	55	oveq2d 7376	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∧ (𝑛‘𝑥)) = (𝑥 ∧ ( ⊥ ‘𝑥)))
67	66	eqeq1d 2739	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∧ (𝑛‘𝑥)) = 0 ↔ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ))
68	63, 65, 67	3anbi123d 1439	. . . . 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 3202	. . . 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 3479	. . 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 629	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  dom cdm 5624  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  lecple 17218  occoc 17219  Posetcpo 18264  lubclub 18266  glbcglb 18267  joincjn 18268  meetcmee 18269  0.cp0 18378  1.cp1 18379  OPcops 39632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5634  df-iota 6448  df-fv 6500  df-ov 7363  df-oposet 39636

This theorem is referenced by:  opposet  39641  oposlem  39642  op01dm  39643