Theorem isopos 40205

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 6877	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2		isopos.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2814	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4		fveq2 6877	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5		isopos.e	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝐾)
6	4, 5	eqtr4di 2814	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
7	6	dmeqd 5887	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
8	3, 7	eleq12d 2855	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9		fveq2 6877	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10		isopos.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
11	9, 10	eqtr4di 2814	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
12	11	dmeqd 5887	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
13	3, 12	eleq12d 2855	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
14	8, 13	anbi12d 644	. . . 4 ⊢ (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)))
15		fveq2 6877	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16		isopos.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
17	15, 16	eqtr4di 2814	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = ⊥ )
18	17	eqeq2d 2772	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ⊥ ))
19	3	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → ((𝑛‘𝑥) ∈ (Base‘𝑝) ↔ (𝑛‘𝑥) ∈ 𝐵))
20		fveq2 6877	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21		isopos.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
22	20, 21	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
23	22	breqd 5114	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦 ↔ 𝑥 ≤ 𝑦))
24	22	breqd 5114	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → ((𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥) ↔ (𝑛‘𝑦) ≤ (𝑛‘𝑥)))
25	23, 24	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → ((𝑥(le‘𝑝)𝑦 → (𝑛‘𝑦)(le‘𝑝)(𝑛‘𝑥)) ↔ (𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥))))
26	19, 25	3anbi13d 1466	. . . . . . . . 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 6877	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28		isopos.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
29	27, 28	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = ∨ )
30	29	oveqd 7429	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛‘𝑥)) = (𝑥 ∨ (𝑛‘𝑥)))
31		fveq2 6877	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32		isopos.u	. . . . . . . . . . 11 ⊢ 1 = (1.‘𝐾)
33	31, 32	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
34	30, 33	eqeq12d 2777	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛‘𝑥)) = (1.‘𝑝) ↔ (𝑥 ∨ (𝑛‘𝑥)) = 1 ))
35		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36		isopos.m	. . . . . . . . . . . 12 ⊢ ∧ = (meet‘𝐾)
37	35, 36	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = ∧ )
38	37	oveqd 7429	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (𝑥 ∧ (𝑛‘𝑥)))
39		fveq2 6877	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40		isopos.f	. . . . . . . . . . 11 ⊢ 0 = (0.‘𝐾)
41	39, 40	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
42	38, 41	eqeq12d 2777	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛‘𝑥)) = (0.‘𝑝) ↔ (𝑥 ∧ (𝑛‘𝑥)) = 0 ))
43	26, 34, 42	3anbi123d 1464	. . . . . . . 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 3335	. . . . . . 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 3335	. . . . . 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 644	. . . . 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 1954	. . . 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 644	. . 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 40201	. . 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 474	. 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 1111	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ↔ (𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)))
53	52	bicomi 227	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) ↔ (𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))
54	16	fvexi 6891	. . . 4 ⊢ ⊥ ∈ V
55		fveq1 6876	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘𝑥) = ( ⊥ ‘𝑥))
56	55	eleq1d 2846	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑥) ∈ 𝐵))
57		id 23	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → 𝑛 = ⊥ )
58	57, 55	fveq12d 6884	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘(𝑛‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑥)))
59	58	eqeq1d 2763	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘(𝑛‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
60		fveq1 6876	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → (𝑛‘𝑦) = ( ⊥ ‘𝑦))
61	60, 55	breq12d 5116	. . . . . . . 8 ⊢ (𝑛 = ⊥ → ((𝑛‘𝑦) ≤ (𝑛‘𝑥) ↔ ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥)))
62	61	imbi2d 343	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑥 ≤ 𝑦 → (𝑛‘𝑦) ≤ (𝑛‘𝑥)) ↔ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))))
63	56, 59, 62	3anbi123d 1464	. . . . . 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 7428	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∨ (𝑛‘𝑥)) = (𝑥 ∨ ( ⊥ ‘𝑥)))
65	64	eqeq1d 2763	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∨ (𝑛‘𝑥)) = 1 ↔ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ))
66	55	oveq2d 7428	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∧ (𝑛‘𝑥)) = (𝑥 ∧ ( ⊥ ‘𝑥)))
67	66	eqeq1d 2763	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∧ (𝑛‘𝑥)) = 0 ↔ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ))
68	63, 65, 67	3anbi123d 1464	. . . . 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 3227	. . . 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 3499	. . 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 640	. 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 302	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 )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  dom cdm 5651  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  lecple 17415  occoc 17416  Posetcpo 18461  lubclub 18463  glbcglb 18464  joincjn 18465  meetcmee 18466  0.cp0 18575  1.cp1 18576  OPcops 40197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5661  df-iota 6487  df-fv 6539  df-ov 7415  df-oposet 40201

This theorem is used by:  opposet  40206  oposlem  40207  op01dm  40208