Theorem caucvgprlemcl 7677

Description: Lemma for caucvgpr 7683. 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 5517	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
5	4	breq2d 4017	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘𝑎)))
6	5	cbvralv 2705	. . . . 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 3780	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
10	9	eceq1d 6573	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
11	10	fveq2d 5521	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
12	11	oveq2d 5893	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
13	12, 4	breq12d 4018	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)))
14	13	cbvrexv 2706	. . . . . . . 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 2724	. . . . . 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 5895	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
18	17	breq1d 4015	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
19	18	cbvrexv 2706	. . . . . . . 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 2724	. . . . . 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 3785	. . . . 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 2198	. . . 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 7669	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
25		ssrab2 3242	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ⊆ Q
26		nqex 7364	. . . . . . 7 ⊢ Q ∈ V
27	26	elpw2 4159	. . . . . 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 3242	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q
30	26	elpw2 4159	. . . . . 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 4660	. . . . 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 2250	. . 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 7674	. . 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 4008	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑛 <N 𝑘 ↔ 𝑐 <N 𝑘))
38		fveq2 5517	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → (𝐹‘𝑛) = (𝐹‘𝑐))
39		opeq1 3780	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ⟨𝑛, 1o⟩ = ⟨𝑐, 1o⟩)
40	39	eceq1d 6573	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → [⟨𝑛, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
41	40	fveq2d 5521	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
42	41	oveq2d 5893	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
43	38, 42	breq12d 4018	. . . . . . . 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 5895	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
45	44	breq2d 4017	. . . . . . . 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 4009	. . . . . . 7 ⊢ (𝑘 = 𝑑 → (𝑐 <N 𝑘 ↔ 𝑐 <N 𝑑))
49		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑘 = 𝑑 → (𝐹‘𝑘) = (𝐹‘𝑑))
50	49	oveq1d 5892	. . . . . . . . 9 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) = ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
51	50	breq2d 4017	. . . . . . . 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 4015	. . . . . . . 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 2718	. . . . 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 7675	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
58	1, 2, 7, 23	caucvgprlemloc 7676	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
59	36, 57, 58	3jca 1177	. 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 7477	. 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 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3131  𝒫 cpw 3577  ⟨cop 3597   class class class wbr 4005   × cxp 4626  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  1^st c1st 6141  2^nd c2nd 6142  1_oc1o 6412  [cec 6535  Ncnpi 7273   <_N clti 7276   ~_Q ceq 7280  Qcnq 7281   +_Q cplq 7283  *_Qcrq 7285   <_Q cltq 7286  Pcnp 7292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467

This theorem is referenced by:  caucvgprlemladdfu  7678  caucvgprlemladdrl  7679  caucvgprlem1  7680  caucvgprlem2  7681  caucvgpr  7683