Theorem caucvgprlemcl 6774

Description: Lemma for caucvgpr 6780. 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 5178	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
5	4	breq2d 3776	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘𝑎)))
6	5	cbvralv 2533	. . . . 5 ⊢ (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) ↔ ∀𝑎 ∈ N 𝐴 <Q (𝐹‘𝑎))
7	3, 6	sylib 127	. . . 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 3549	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
10	9	eceq1d 6142	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
11	10	fveq2d 5182	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
12	11	oveq2d 5528	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
13	12, 4	breq12d 3777	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑎)))
14	13	cbvrexv 2534	. . . . . . . 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 2547	. . . . . 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 5530	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
18	17	breq1d 3774	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢))
19	18	cbvrexv 2534	. . . . . . . 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 2547	. . . . . 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 3554	. . . . 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 2060	. . . 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 6766	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
25		ssrab2 3025	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)} ⊆ Q
26		nqex 6461	. . . . . . 7 ⊢ Q ∈ V
27	26	elpw2 3911	. . . . . 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 134	. . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ 𝒫 Q
29		ssrab2 3025	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ⊆ Q
30	26	elpw2 3911	. . . . . 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 134	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q
32		opelxpi 4376	. . . . 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 402	. . . 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 2110	. . 3 ⊢ 𝐿 ∈ (𝒫 Q × 𝒫 Q)
35	24, 34	jctil 295	. 2 ⊢ (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))))
36	1, 2, 7, 23	caucvgprlemrnd 6771	. . 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 3767	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑛 <N 𝑘 ↔ 𝑐 <N 𝑘))
38		fveq2 5178	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → (𝐹‘𝑛) = (𝐹‘𝑐))
39		opeq1 3549	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ⟨𝑛, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
40	39	eceq1d 6142	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
41	40	fveq2d 5182	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
42	41	oveq2d 5528	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
43	38, 42	breq12d 3777	. . . . . . . 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 5530	. . . . . . . . 9 ⊢ (𝑛 = 𝑐 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹‘𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
45	44	breq2d 3776	. . . . . . . 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 442	. . . . . . 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 223	. . . . . 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 3768	. . . . . . 7 ⊢ (𝑘 = 𝑑 → (𝑐 <N 𝑘 ↔ 𝑐 <N 𝑑))
49		fveq2 5178	. . . . . . . . . 10 ⊢ (𝑘 = 𝑑 → (𝐹‘𝑘) = (𝐹‘𝑑))
50	49	oveq1d 5527	. . . . . . . . 9 ⊢ (𝑘 = 𝑑 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) = ((𝐹‘𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
51	50	breq2d 3776	. . . . . . . 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 3774	. . . . . . . 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 442	. . . . . . 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 223	. . . . . 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 2541	. . . . 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 127	. . . 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 6772	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
58	1, 2, 7, 23	caucvgprlemloc 6773	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
59	36, 57, 58	3jca 1084	. 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 6574	. 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 394	1 ⊢ (𝜑 → 𝐿 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310   ⊆ wss 2917  𝒫 cpw 3359  ⟨cop 3378   class class class wbr 3764   × cxp 4343  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  caucvgprlemladdfu  6775  caucvgprlemladdrl  6776  caucvgprlem1  6777  caucvgprlem2  6778  caucvgpr  6780