Theorem isopos 37642

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 6842	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2		isopos.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2794	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4		fveq2 6842	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5		isopos.e	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝐾)
6	4, 5	eqtr4di 2794	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
7	6	dmeqd 5861	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
8	3, 7	eleq12d 2832	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9		fveq2 6842	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10		isopos.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
11	9, 10	eqtr4di 2794	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
12	11	dmeqd 5861	. . . . . 6 ⊢ (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
13	3, 12	eleq12d 2832	. . . . 5 ⊢ (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
14	8, 13	anbi12d 631	. . . 4 ⊢ (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)))
15		fveq2 6842	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16		isopos.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
17	15, 16	eqtr4di 2794	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = ⊥ )
18	17	eqeq2d 2747	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ⊥ ))
19	3	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → ((𝑛‘𝑥) ∈ (Base‘𝑝) ↔ (𝑛‘𝑥) ∈ 𝐵))
20		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21		isopos.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
22	20, 21	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
23	22	breqd 5116	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦 ↔ 𝑥 ≤ 𝑦))
24	22	breqd 5116	. . . . . . . . . . 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 1438	. . . . . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28		isopos.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
29	27, 28	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = ∨ )
30	29	oveqd 7374	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛‘𝑥)) = (𝑥 ∨ (𝑛‘𝑥)))
31		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32		isopos.u	. . . . . . . . . . 11 ⊢ 1 = (1.‘𝐾)
33	31, 32	eqtr4di 2794	. . . . . . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36		isopos.m	. . . . . . . . . . . 12 ⊢ ∧ = (meet‘𝐾)
37	35, 36	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = ∧ )
38	37	oveqd 7374	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛‘𝑥)) = (𝑥 ∧ (𝑛‘𝑥)))
39		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40		isopos.f	. . . . . . . . . . 11 ⊢ 0 = (0.‘𝐾)
41	39, 40	eqtr4di 2794	. . . . . . . . . 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 1436	. . . . . . . 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 3319	. . . . . . 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 3319	. . . . . 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 631	. . . . 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 1924	. . . 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 631	. . 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 37638	. . 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 3648	. 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 469	. 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 223	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) ↔ (𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))
54	16	fvexi 6856	. . . 4 ⊢ ⊥ ∈ V
55		fveq1 6841	. . . . . . . 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 6849	. . . . . . . 8 ⊢ (𝑛 = ⊥ → (𝑛‘(𝑛‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑥)))
59	58	eqeq1d 2738	. . . . . . 7 ⊢ (𝑛 = ⊥ → ((𝑛‘(𝑛‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
60		fveq1 6841	. . . . . . . . 9 ⊢ (𝑛 = ⊥ → (𝑛‘𝑦) = ( ⊥ ‘𝑦))
61	60, 55	breq12d 5118	. . . . . . . 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 1436	. . . . . 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 7373	. . . . . . 7 ⊢ (𝑛 = ⊥ → (𝑥 ∨ (𝑛‘𝑥)) = (𝑥 ∨ ( ⊥ ‘𝑥)))
65	64	eqeq1d 2738	. . . . . 6 ⊢ (𝑛 = ⊥ → ((𝑥 ∨ (𝑛‘𝑥)) = 1 ↔ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ))
66	55	oveq2d 7373	. . . . . . 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 1436	. . . . 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 3212	. . . 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 3494	. . 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 627	. 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 298	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064   class class class wbr 5105  dom cdm 5633  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  occoc 17141  Posetcpo 18196  lubclub 18198  glbcglb 18199  joincjn 18200  meetcmee 18201  0.cp0 18312  1.cp1 18313  OPcops 37634

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-dm 5643  df-iota 6448  df-fv 6504  df-ov 7360  df-oposet 37638

This theorem is referenced by:  opposet  37643  oposlem  37644  op01dm  37645