Theorem caucvgprlemcl 7743

Description: Lemma for caucvgpr 7749. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩

Assertion

Ref	Expression
caucvgprlemcl	⊢ (𝜑 → 𝐿 ∈ P)

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙   𝑢,𝐹,𝑗   𝑛,𝐹,𝑘   𝑗,𝑘,𝐿   𝑘,𝑛

Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlemcl

Dummy variables 𝑠 𝑎 𝑐 𝑑 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . 4 ⊢ (𝜑 → 𝐹:N⟶Q)
2		caucvgpr.cau	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
3		caucvgpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
4		fveq2 5558	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
5	4	breq2d 4045	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘𝑎)))
6	5	cbvralv 2729	. . . . 5 ⊢ (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) ↔ ∀𝑎 ∈ N 𝐴 <Q (𝐹‘𝑎))
7	3, 6	sylib 122	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ N 𝐴 <Q (𝐹‘𝑎))
8		caucvgpr.lim	. . . . 5 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
9		opeq1 3808	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
10	9	eceq1d 6628	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
11	10	fveq2d 5562	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
12	11	oveq2d 5938	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
13	12, 4	breq12d 4046	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)))
14	13	cbvrexv 2730	. . . . . . . 8 ⊢ (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎))
15	14	a1i 9	. . . . . . 7 ⊢ (𝑙 ∈ Q → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)))
16	15	rabbiia 2748	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} = {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)}
17	4, 11	oveq12d 5940	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
18	17	breq1d 4043	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
19	18	cbvrexv 2730	. . . . . . . 8 ⊢ (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑎 ∈ N ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢)
20	19	a1i 9	. . . . . . 7 ⊢ (𝑢 ∈ Q → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑎 ∈ N ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
21	20	rabbiia 2748	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑎 ∈ N ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}
22	16, 21	opeq12i 3813	. . . . 5 ⊢ ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)}, {𝑢 ∈ Q ∣ ∃𝑎 ∈ N ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩
23	8, 22	eqtri 2217	. . . 4 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)}, {𝑢 ∈ Q ∣ ∃𝑎 ∈ N ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩
24	1, 2, 7, 23	caucvgprlemm 7735	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
25		ssrab2 3268	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ⊆ Q
26		nqex 7430	. . . . . . 7 ⊢ Q ∈ V
27	26	elpw2 4190	. . . . . 6 ⊢ ({𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ 𝒫 Q ↔ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ⊆ Q)
28	25, 27	mpbir 146	. . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ 𝒫 Q
29		ssrab2 3268	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q
30	26	elpw2 4190	. . . . . 6 ⊢ ({𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q ↔ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q)
31	29, 30	mpbir 146	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q
32		opelxpi 4695	. . . . 5 ⊢ (({𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ 𝒫 Q ∧ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q) → ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q))
33	28, 31, 32	mp2an 426	. . . 4 ⊢ ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q)
34	8, 33	eqeltri 2269	. . 3 ⊢ 𝐿 ∈ (𝒫 Q × 𝒫 Q)
35	24, 34	jctil 312	. 2 ⊢ (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))))
36	1, 2, 7, 23	caucvgprlemrnd 7740	. . 3 ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))))
37		breq1 4036	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑛 <N 𝑘 ↔ 𝑐 <N 𝑘))
38		fveq2 5558	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → (𝐹‘𝑛) = (𝐹‘𝑐))
39		opeq1 3808	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ⟨𝑛, 1o⟩ = ⟨𝑐, 1o⟩)
40	39	eceq1d 6628	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → [⟨𝑛, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
41	40	fveq2d 5562	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
42	41	oveq2d 5938	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
43	38, 42	breq12d 4046	. . . . . . . 8 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹‘𝑐) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
44	38, 41	oveq12d 5940	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
45	44	breq2d 4045	. . . . . . . 8 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹‘𝑘) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
46	43, 45	anbi12d 473	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q ))) ↔ ((𝐹‘𝑐) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
47	37, 46	imbi12d 234	. . . . . 6 ⊢ (𝑛 = 𝑐 → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) ↔ (𝑐 <N 𝑘 → ((𝐹‘𝑐) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))))
48		breq2 4037	. . . . . . 7 ⊢ (𝑘 = 𝑑 → (𝑐 <N 𝑘 ↔ 𝑐 <N 𝑑))
49		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑘 = 𝑑 → (𝐹‘𝑘) = (𝐹‘𝑑))
50	49	oveq1d 5937	. . . . . . . . 9 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) = ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
51	50	breq2d 4045	. . . . . . . 8 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑐) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ↔ (𝐹‘𝑐) <Q ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
52	49	breq1d 4043	. . . . . . . 8 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑘) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ↔ (𝐹‘𝑑) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
53	51, 52	anbi12d 473	. . . . . . 7 ⊢ (𝑘 = 𝑑 → (((𝐹‘𝑐) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))) ↔ ((𝐹‘𝑐) <Q ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑑) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
54	48, 53	imbi12d 234	. . . . . 6 ⊢ (𝑘 = 𝑑 → ((𝑐 <N 𝑘 → ((𝐹‘𝑐) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))) ↔ (𝑐 <N 𝑑 → ((𝐹‘𝑐) <Q ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑑) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))))
55	47, 54	cbvral2v 2742	. . . . 5 ⊢ (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) ↔ ∀𝑐 ∈ N ∀𝑑 ∈ N (𝑐 <N 𝑑 → ((𝐹‘𝑐) <Q ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑑) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
56	2, 55	sylib 122	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ N ∀𝑑 ∈ N (𝑐 <N 𝑑 → ((𝐹‘𝑐) <Q ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∧ (𝐹‘𝑑) <Q ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))))
57	1, 56, 7, 23	caucvgprlemdisj 7741	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
58	1, 2, 7, 23	caucvgprlemloc 7742	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
59	36, 57, 58	3jca 1179	. 2 ⊢ (𝜑 → ((∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))) ∧ ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) ∧ ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))))
60		elnp1st2nd 7543	. 2 ⊢ (𝐿 ∈ P ↔ ((𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))) ∧ ((∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))) ∧ ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) ∧ ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))))
61	35, 59, 60	sylanbrc 417	1 ⊢ (𝜑 → 𝐿 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479   ⊆ wss 3157  𝒫 cpw 3605  ⟨cop 3625   class class class wbr 4033   × cxp 4661  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  1^st c1st 6196  2^nd c2nd 6197  1_oc1o 6467  [cec 6590  Ncnpi 7339   <_N clti 7342   ~_Q ceq 7346  Qcnq 7347   +_Q cplq 7349  *_Qcrq 7351   <_Q cltq 7352  Pcnp 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533

This theorem is referenced by:  caucvgprlemladdfu  7744  caucvgprlemladdrl  7745  caucvgprlem1  7746  caucvgprlem2  7747  caucvgpr  7749