Theorem caucvgprlemcl 7617

Description: Lemma for caucvgpr 7623. 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 5486	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
5	4	breq2d 3994	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘𝑎)))
6	5	cbvralv 2692	. . . . 5 ⊢ (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) ↔ ∀𝑎 ∈ N 𝐴 <Q (𝐹‘𝑎))
7	3, 6	sylib 121	. . . 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 3758	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
10	9	eceq1d 6537	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
11	10	fveq2d 5490	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
12	11	oveq2d 5858	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
13	12, 4	breq12d 3995	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)))
14	13	cbvrexv 2693	. . . . . . . 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 2711	. . . . . 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 5860	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
18	17	breq1d 3992	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢))
19	18	cbvrexv 2693	. . . . . . . 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 2711	. . . . . 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 3763	. . . . 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 2186	. . . 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 7609	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
25		ssrab2 3227	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ⊆ Q
26		nqex 7304	. . . . . . 7 ⊢ Q ∈ V
27	26	elpw2 4136	. . . . . 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 145	. . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ 𝒫 Q
29		ssrab2 3227	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ⊆ Q
30	26	elpw2 4136	. . . . . 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 145	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q
32		opelxpi 4636	. . . . 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 423	. . . 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 2239	. . 3 ⊢ 𝐿 ∈ (𝒫 Q × 𝒫 Q)
35	24, 34	jctil 310	. 2 ⊢ (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))))
36	1, 2, 7, 23	caucvgprlemrnd 7614	. . 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 3985	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑛 <N 𝑘 ↔ 𝑐 <N 𝑘))
38		fveq2 5486	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → (𝐹‘𝑛) = (𝐹‘𝑐))
39		opeq1 3758	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ⟨𝑛, 1o⟩ = ⟨𝑐, 1o⟩)
40	39	eceq1d 6537	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → [⟨𝑛, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
41	40	fveq2d 5490	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
42	41	oveq2d 5858	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
43	38, 42	breq12d 3995	. . . . . . . 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 5860	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
45	44	breq2d 3994	. . . . . . . 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 465	. . . . . . 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 233	. . . . . 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 3986	. . . . . . 7 ⊢ (𝑘 = 𝑑 → (𝑐 <N 𝑘 ↔ 𝑐 <N 𝑑))
49		fveq2 5486	. . . . . . . . . 10 ⊢ (𝑘 = 𝑑 → (𝐹‘𝑘) = (𝐹‘𝑑))
50	49	oveq1d 5857	. . . . . . . . 9 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) = ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
51	50	breq2d 3994	. . . . . . . 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 3992	. . . . . . . 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 465	. . . . . . 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 233	. . . . . 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 2705	. . . . 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 121	. . . 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 7615	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
58	1, 2, 7, 23	caucvgprlemloc 7616	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
59	36, 57, 58	3jca 1167	. 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 7417	. 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 414	1 ⊢ (𝜑 → 𝐿 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  {crab 2448   ⊆ wss 3116  𝒫 cpw 3559  ⟨cop 3579   class class class wbr 3982   × cxp 4602  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  2^nd c2nd 6107  1_oc1o 6377  [cec 6499  Ncnpi 7213   <_N clti 7216   ~_Q ceq 7220  Qcnq 7221   +_Q cplq 7223  *_Qcrq 7225   <_Q cltq 7226  Pcnp 7232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407

This theorem is referenced by:  caucvgprlemladdfu  7618  caucvgprlemladdrl  7619  caucvgprlem1  7620  caucvgprlem2  7621  caucvgpr  7623