Theorem caucvgprlemm 6858

Description: Lemma for caucvgpr 6872. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-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
caucvgprlemm	⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Distinct variable groups:   𝐴,𝑗,𝑠   𝑗,𝐹,𝑙   𝐹,𝑟   𝑢,𝐹,𝑗   𝐿,𝑟   𝜑,𝑗,𝑠   𝑠,𝑙

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

Proof of Theorem caucvgprlemm

Step	Hyp	Ref	Expression
1		1pi 6505	. . . . 5 ⊢ 1𝑜 ∈ N
2		caucvgpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
3		fveq2 5198	. . . . . . 7 ⊢ (𝑗 = 1𝑜 → (𝐹‘𝑗) = (𝐹‘1𝑜))
4	3	breq2d 3797	. . . . . 6 ⊢ (𝑗 = 1𝑜 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘1𝑜)))
5	4	rspcv 2697	. . . . 5 ⊢ (1𝑜 ∈ N → (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) → 𝐴 <Q (𝐹‘1𝑜)))
6	1, 2, 5	mpsyl 64	. . . 4 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1𝑜))
7		ltrelnq 6555	. . . . . 6 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4410	. . . . 5 ⊢ (𝐴 <Q (𝐹‘1𝑜) → (𝐴 ∈ Q ∧ (𝐹‘1𝑜) ∈ Q))
9	8	simpld 110	. . . 4 ⊢ (𝐴 <Q (𝐹‘1𝑜) → 𝐴 ∈ Q)
10		halfnqq 6600	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
11	6, 9, 10	3syl 17	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
12		simplr 496	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
13		archrecnq 6853	. . . . . . . 8 ⊢ (𝑠 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
14	12, 13	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
15		simpr 108	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠)
16		simplr 496	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑗 ∈ N)
17		nnnq 6612	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ N → [⟨𝑗, 1𝑜⟩] ~Q ∈ Q)
18		recclnq 6582	. . . . . . . . . . . . . 14 ⊢ ([⟨𝑗, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
19	16, 17, 18	3syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
20	12	ad2antrr 471	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑠 ∈ Q)
21		ltanqg 6590	. . . . . . . . . . . . 13 ⊢ (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q ∧ 𝑠 ∈ Q) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
22	19, 20, 20, 21	syl3anc 1169	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
23	15, 22	mpbid 145	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝑠 +Q 𝑠))
24		simpllr 500	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
25	23, 24	breqtrd 3809	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴)
26		rsp 2411	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) → (𝑗 ∈ N → 𝐴 <Q (𝐹‘𝑗)))
27	2, 26	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ N → 𝐴 <Q (𝐹‘𝑗)))
28	27	ad4antr 477	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑗 ∈ N → 𝐴 <Q (𝐹‘𝑗)))
29	16, 28	mpd 13	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝐴 <Q (𝐹‘𝑗))
30		ltsonq 6588	. . . . . . . . . . 11 ⊢ <Q Or Q
31	30, 7	sotri 4740	. . . . . . . . . 10 ⊢ (((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝐴 ∧ 𝐴 <Q (𝐹‘𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗))
32	25, 29, 31	syl2anc 403	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗))
33	32	ex 113	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
34	33	reximdva 2463	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (∃𝑗 ∈ N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑠 → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
35	14, 34	mpd 13	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗))
36		oveq1 5539	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
37	36	breq1d 3795	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
38	37	rexbidv 2369	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
39		caucvgpr.lim	. . . . . . . . 9 ⊢ 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}⟩
40	39	fveq2i 5201	. . . . . . . 8 ⊢ (1st ‘𝐿) = (1st ‘⟨{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}⟩)
41		nqex 6553	. . . . . . . . . 10 ⊢ Q ∈ V
42	41	rabex 3922	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
43	41	rabex 3922	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
44	42, 43	op1st 5793	. . . . . . . 8 ⊢ (1st ‘⟨{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 ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}
45	40, 44	eqtri 2101	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}
46	38, 45	elrab2 2751	. . . . . 6 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
47	12, 35, 46	sylanbrc 408	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st ‘𝐿))
48	47	ex 113	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ Q) → ((𝑠 +Q 𝑠) = 𝐴 → 𝑠 ∈ (1st ‘𝐿)))
49	48	reximdva 2463	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿)))
50	11, 49	mpd 13	. 2 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))
51		caucvgpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:N⟶Q)
52	1	a1i 9	. . . . . 6 ⊢ (𝜑 → 1𝑜 ∈ N)
53	51, 52	ffvelrnd 5324	. . . . 5 ⊢ (𝜑 → (𝐹‘1𝑜) ∈ Q)
54		1nq 6556	. . . . 5 ⊢ 1Q ∈ Q
55		addclnq 6565	. . . . 5 ⊢ (((𝐹‘1𝑜) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
56	53, 54, 55	sylancl 404	. . . 4 ⊢ (𝜑 → ((𝐹‘1𝑜) +Q 1Q) ∈ Q)
57		addclnq 6565	. . . 4 ⊢ ((((𝐹‘1𝑜) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q)
58	56, 54, 57	sylancl 404	. . 3 ⊢ (𝜑 → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q)
59		df-1nqqs 6541	. . . . . . . . 9 ⊢ 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
60	59	fveq2i 5201	. . . . . . . 8 ⊢ (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
61		rec1nq 6585	. . . . . . . 8 ⊢ (*Q‘1Q) = 1Q
62	60, 61	eqtr3i 2103	. . . . . . 7 ⊢ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
63	62	oveq2i 5543	. . . . . 6 ⊢ ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q 1Q)
64		ltaddnq 6597	. . . . . . 7 ⊢ ((((𝐹‘1𝑜) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
65	56, 54, 64	sylancl 404	. . . . . 6 ⊢ (𝜑 → ((𝐹‘1𝑜) +Q 1Q) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
66	63, 65	syl5eqbr 3818	. . . . 5 ⊢ (𝜑 → ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
67		opeq1 3570	. . . . . . . . . 10 ⊢ (𝑗 = 1𝑜 → ⟨𝑗, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
68	67	eceq1d 6165	. . . . . . . . 9 ⊢ (𝑗 = 1𝑜 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
69	68	fveq2d 5202	. . . . . . . 8 ⊢ (𝑗 = 1𝑜 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
70	3, 69	oveq12d 5550	. . . . . . 7 ⊢ (𝑗 = 1𝑜 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
71	70	breq1d 3795	. . . . . 6 ⊢ (𝑗 = 1𝑜 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ↔ ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
72	71	rspcev 2701	. . . . 5 ⊢ ((1𝑜 ∈ N ∧ ((𝐹‘1𝑜) +Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
73	52, 66, 72	syl2anc 403	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q))
74		breq2 3789	. . . . . 6 ⊢ (𝑢 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
75	74	rexbidv 2369	. . . . 5 ⊢ (𝑢 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
76	39	fveq2i 5201	. . . . . 6 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{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}⟩)
77	42, 43	op2nd 5794	. . . . . 6 ⊢ (2nd ‘⟨{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}⟩) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
78	76, 77	eqtri 2101	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
79	75, 78	elrab2 2751	. . . 4 ⊢ ((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿) ↔ ((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (((𝐹‘1𝑜) +Q 1Q) +Q 1Q)))
80	58, 73, 79	sylanbrc 408	. . 3 ⊢ (𝜑 → (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿))
81		eleq1 2141	. . . 4 ⊢ (𝑟 = (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
82	81	rspcev 2701	. . 3 ⊢ (((((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1𝑜) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
83	58, 80, 82	syl2anc 403	. 2 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
84	50, 83	jca 300	1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  {crab 2352  ⟨cop 3401   class class class wbr 3785  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  1_𝑜c1o 6017  [cec 6127  Ncnpi 6462   <_N clti 6465   ~_Q ceq 6469  Qcnq 6470  1_Qc1q 6471   +_Q cplq 6472  *_Qcrq 6474   <_Q cltq 6475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543

This theorem is referenced by:  caucvgprlemcl  6866