Theorem caucvgprlemcl 7802

Description: Lemma for caucvgpr 7808. 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 5586	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
5	4	breq2d 4060	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘𝑎)))
6	5	cbvralv 2739	. . . . 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 3822	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
10	9	eceq1d 6666	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
11	10	fveq2d 5590	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
12	11	oveq2d 5970	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
13	12, 4	breq12d 4061	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)))
14	13	cbvrexv 2740	. . . . . . . 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 2758	. . . . . 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 5972	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
18	17	breq1d 4058	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
19	18	cbvrexv 2740	. . . . . . . 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 2758	. . . . . 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 3827	. . . . 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 2227	. . . 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 7794	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
25		ssrab2 3280	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ⊆ Q
26		nqex 7489	. . . . . . 7 ⊢ Q ∈ V
27	26	elpw2 4206	. . . . . 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 3280	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q
30	26	elpw2 4206	. . . . . 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 4712	. . . . 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 2279	. . 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 7799	. . 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 4051	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑛 <N 𝑘 ↔ 𝑐 <N 𝑘))
38		fveq2 5586	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → (𝐹‘𝑛) = (𝐹‘𝑐))
39		opeq1 3822	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ⟨𝑛, 1o⟩ = ⟨𝑐, 1o⟩)
40	39	eceq1d 6666	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → [⟨𝑛, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
41	40	fveq2d 5590	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
42	41	oveq2d 5970	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
43	38, 42	breq12d 4061	. . . . . . . 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 5972	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
45	44	breq2d 4060	. . . . . . . 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 4052	. . . . . . 7 ⊢ (𝑘 = 𝑑 → (𝑐 <N 𝑘 ↔ 𝑐 <N 𝑑))
49		fveq2 5586	. . . . . . . . . 10 ⊢ (𝑘 = 𝑑 → (𝐹‘𝑘) = (𝐹‘𝑑))
50	49	oveq1d 5969	. . . . . . . . 9 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) = ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
51	50	breq2d 4060	. . . . . . . 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 4058	. . . . . . . 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 2752	. . . . 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 7800	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
58	1, 2, 7, 23	caucvgprlemloc 7801	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
59	36, 57, 58	3jca 1180	. 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 7602	. 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 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489   ⊆ wss 3168  𝒫 cpw 3618  ⟨cop 3638   class class class wbr 4048   × cxp 4678  ⟶wf 5273  ‘cfv 5277  (class class class)co 5954  1^st c1st 6234  2^nd c2nd 6235  1_oc1o 6505  [cec 6628  Ncnpi 7398   <_N clti 7401   ~_Q ceq 7405  Qcnq 7406   +_Q cplq 7408  *_Qcrq 7410   <_Q cltq 7411  Pcnp 7417

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-eprel 4341  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-1o 6512  df-oadd 6516  df-omul 6517  df-er 6630  df-ec 6632  df-qs 6636  df-ni 7430  df-pli 7431  df-mi 7432  df-lti 7433  df-plpq 7470  df-mpq 7471  df-enq 7473  df-nqqs 7474  df-plqqs 7475  df-mqqs 7476  df-1nqqs 7477  df-rq 7478  df-ltnqqs 7479  df-inp 7592

This theorem is referenced by:  caucvgprlemladdfu  7803  caucvgprlemladdrl  7804  caucvgprlem1  7805  caucvgprlem2  7806  caucvgpr  7808