Theorem caucvgprlemm 7630

Description: Lemma for caucvgpr 7644. 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		fveq2 5496	. . . . . 6 ⊢ (𝑗 = 1o → (𝐹‘𝑗) = (𝐹‘1o))
2	1	breq2d 4001	. . . . 5 ⊢ (𝑗 = 1o → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘1o)))
3		caucvgpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
4		1pi 7277	. . . . . 6 ⊢ 1o ∈ N
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → 1o ∈ N)
6	2, 3, 5	rspcdva 2839	. . . 4 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1o))
7		ltrelnq 7327	. . . . . 6 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4663	. . . . 5 ⊢ (𝐴 <Q (𝐹‘1o) → (𝐴 ∈ Q ∧ (𝐹‘1o) ∈ Q))
9	8	simpld 111	. . . 4 ⊢ (𝐴 <Q (𝐹‘1o) → 𝐴 ∈ Q)
10		halfnqq 7372	. . . 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 525	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
13		archrecnq 7625	. . . . . . . 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 109	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠)
16		simplr 525	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑗 ∈ N)
17		nnnq 7384	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
18		recclnq 7354	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑠 ∈ Q)
21		ltanqg 7362	. . . . . . . . . . . . 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 1233	. . . . . . . . . . . 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 146	. . . . . . . . . . 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 529	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
25	23, 24	breqtrd 4015	. . . . . . . . . 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 2517	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) → (𝑗 ∈ N → 𝐴 <Q (𝐹‘𝑗)))
27	3, 26	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ N → 𝐴 <Q (𝐹‘𝑗)))
28	27	ad4antr 491	. . . . . . . . . . 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 7360	. . . . . . . . . . 11 ⊢ <Q Or Q
31	30, 7	sotri 5006	. . . . . . . . . 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 409	. . . . . . . . 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 114	. . . . . . . 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 2572	. . . . . . 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 5860	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
37	36	breq1d 3999	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
38	37	rexbidv 2471	. . . . . . 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 5499	. . . . . . . 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 7325	. . . . . . . . . 10 ⊢ Q ∈ V
42	41	rabex 4133	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
43	41	rabex 4133	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
44	42, 43	op1st 6125	. . . . . . . 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 2191	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
46	38, 45	elrab2 2889	. . . . . 6 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
47	12, 35, 46	sylanbrc 415	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st ‘𝐿))
48	47	ex 114	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ Q) → ((𝑠 +Q 𝑠) = 𝐴 → 𝑠 ∈ (1st ‘𝐿)))
49	48	reximdva 2572	. . 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	51, 5	ffvelrnd 5632	. . . . 5 ⊢ (𝜑 → (𝐹‘1o) ∈ Q)
53		1nq 7328	. . . . 5 ⊢ 1Q ∈ Q
54		addclnq 7337	. . . . 5 ⊢ (((𝐹‘1o) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1o) +Q 1Q) ∈ Q)
55	52, 53, 54	sylancl 411	. . . 4 ⊢ (𝜑 → ((𝐹‘1o) +Q 1Q) ∈ Q)
56		addclnq 7337	. . . 4 ⊢ ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
57	55, 53, 56	sylancl 411	. . 3 ⊢ (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
58		df-1nqqs 7313	. . . . . . . . 9 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
59	58	fveq2i 5499	. . . . . . . 8 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
60		rec1nq 7357	. . . . . . . 8 ⊢ (*Q‘1Q) = 1Q
61	59, 60	eqtr3i 2193	. . . . . . 7 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
62	61	oveq2i 5864	. . . . . 6 ⊢ ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) = ((𝐹‘1o) +Q 1Q)
63		ltaddnq 7369	. . . . . . 7 ⊢ ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
64	55, 53, 63	sylancl 411	. . . . . 6 ⊢ (𝜑 → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
65	62, 64	eqbrtrid 4024	. . . . 5 ⊢ (𝜑 → ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
66		opeq1 3765	. . . . . . . . . 10 ⊢ (𝑗 = 1o → ⟨𝑗, 1o⟩ = ⟨1o, 1o⟩)
67	66	eceq1d 6549	. . . . . . . . 9 ⊢ (𝑗 = 1o → [⟨𝑗, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
68	67	fveq2d 5500	. . . . . . . 8 ⊢ (𝑗 = 1o → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
69	1, 68	oveq12d 5871	. . . . . . 7 ⊢ (𝑗 = 1o → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )))
70	69	breq1d 3999	. . . . . 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)))
71	70	rspcev 2834	. . . . 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))
72	5, 65, 71	syl2anc 409	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
73		breq2 3993	. . . . . 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)))
74	73	rexbidv 2471	. . . . 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)))
75	39	fveq2i 5499	. . . . . 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}⟩)
76	42, 43	op2nd 6126	. . . . . 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}
77	75, 76	eqtri 2191	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
78	74, 77	elrab2 2889	. . . 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)))
79	57, 72, 78	sylanbrc 415	. . 3 ⊢ (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿))
80		eleq1 2233	. . . 4 ⊢ (𝑟 = (((𝐹‘1o) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
81	80	rspcev 2834	. . 3 ⊢ (((((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
82	57, 79, 81	syl2anc 409	. 2 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
83	50, 82	jca 304	1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  {crab 2452  ⟨cop 3586   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1^st c1st 6117  2^nd c2nd 6118  1_oc1o 6388  [cec 6511  Ncnpi 7234   <_N clti 7237   ~_Q ceq 7241  Qcnq 7242  1_Qc1q 7243   +_Q cplq 7244  *_Qcrq 7246   <_Q cltq 7247

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  caucvgprlemcl  7638