Theorem caucvgprlemm 7823

Description: Lemma for caucvgpr 7837. 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 5603	. . . . . 6 ⊢ (𝑗 = 1o → (𝐹‘𝑗) = (𝐹‘1o))
2	1	breq2d 4074	. . . . 5 ⊢ (𝑗 = 1o → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘1o)))
3		caucvgpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
4		1pi 7470	. . . . . 6 ⊢ 1o ∈ N
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → 1o ∈ N)
6	2, 3, 5	rspcdva 2892	. . . 4 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1o))
7		ltrelnq 7520	. . . . . 6 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4748	. . . . 5 ⊢ (𝐴 <Q (𝐹‘1o) → (𝐴 ∈ Q ∧ (𝐹‘1o) ∈ Q))
9	8	simpld 112	. . . 4 ⊢ (𝐴 <Q (𝐹‘1o) → 𝐴 ∈ Q)
10		halfnqq 7565	. . . 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 528	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
13		archrecnq 7818	. . . . . . . 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 110	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠)
16		simplr 528	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑗 ∈ N)
17		nnnq 7577	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
18		recclnq 7547	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑠 ∈ Q)
21		ltanqg 7555	. . . . . . . . . . . . 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 1252	. . . . . . . . . . . 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 147	. . . . . . . . . . 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 534	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
25	23, 24	breqtrd 4088	. . . . . . . . . 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 2557	. . . . . . . . . . . . 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 494	. . . . . . . . . . 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 7553	. . . . . . . . . . 11 ⊢ <Q Or Q
31	30, 7	sotri 5100	. . . . . . . . . 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 411	. . . . . . . . 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 115	. . . . . . . 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 2612	. . . . . . 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 5981	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
37	36	breq1d 4072	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
38	37	rexbidv 2511	. . . . . . 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 5606	. . . . . . . 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 7518	. . . . . . . . . 10 ⊢ Q ∈ V
42	41	rabex 4207	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
43	41	rabex 4207	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
44	42, 43	op1st 6262	. . . . . . . 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 2230	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
46	38, 45	elrab2 2942	. . . . . 6 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
47	12, 35, 46	sylanbrc 417	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st ‘𝐿))
48	47	ex 115	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ Q) → ((𝑠 +Q 𝑠) = 𝐴 → 𝑠 ∈ (1st ‘𝐿)))
49	48	reximdva 2612	. . 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	ffvelcdmd 5744	. . . . 5 ⊢ (𝜑 → (𝐹‘1o) ∈ Q)
53		1nq 7521	. . . . 5 ⊢ 1Q ∈ Q
54		addclnq 7530	. . . . 5 ⊢ (((𝐹‘1o) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1o) +Q 1Q) ∈ Q)
55	52, 53, 54	sylancl 413	. . . 4 ⊢ (𝜑 → ((𝐹‘1o) +Q 1Q) ∈ Q)
56		addclnq 7530	. . . 4 ⊢ ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
57	55, 53, 56	sylancl 413	. . 3 ⊢ (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ Q)
58		df-1nqqs 7506	. . . . . . . . 9 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
59	58	fveq2i 5606	. . . . . . . 8 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
60		rec1nq 7550	. . . . . . . 8 ⊢ (*Q‘1Q) = 1Q
61	59, 60	eqtr3i 2232	. . . . . . 7 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
62	61	oveq2i 5985	. . . . . 6 ⊢ ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) = ((𝐹‘1o) +Q 1Q)
63		ltaddnq 7562	. . . . . . 7 ⊢ ((((𝐹‘1o) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
64	55, 53, 63	sylancl 413	. . . . . 6 ⊢ (𝜑 → ((𝐹‘1o) +Q 1Q) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
65	62, 64	eqbrtrid 4097	. . . . 5 ⊢ (𝜑 → ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
66		opeq1 3836	. . . . . . . . . 10 ⊢ (𝑗 = 1o → ⟨𝑗, 1o⟩ = ⟨1o, 1o⟩)
67	66	eceq1d 6686	. . . . . . . . 9 ⊢ (𝑗 = 1o → [⟨𝑗, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
68	67	fveq2d 5607	. . . . . . . 8 ⊢ (𝑗 = 1o → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
69	1, 68	oveq12d 5992	. . . . . . 7 ⊢ (𝑗 = 1o → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )))
70	69	breq1d 4072	. . . . . 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 2887	. . . . 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 411	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
73		breq2 4066	. . . . . 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 2511	. . . . 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 5606	. . . . . 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 6263	. . . . . 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 2230	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
78	74, 77	elrab2 2942	. . . 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 417	. . 3 ⊢ (𝜑 → (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿))
80		eleq1 2272	. . . 4 ⊢ (𝑟 = (((𝐹‘1o) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
81	80	rspcev 2887	. . 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 411	. 2 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
83	50, 82	jca 306	1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  {crab 2492  ⟨cop 3649   class class class wbr 4062  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974  1^st c1st 6254  2^nd c2nd 6255  1_oc1o 6525  [cec 6648  Ncnpi 7427   <_N clti 7430   ~_Q ceq 7434  Qcnq 7435  1_Qc1q 7436   +_Q cplq 7437  *_Qcrq 7439   <_Q cltq 7440

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-1o 6532  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-lti 7462  df-plpq 7499  df-mpq 7500  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-mqqs 7505  df-1nqqs 7506  df-rq 7507  df-ltnqqs 7508

This theorem is referenced by:  caucvgprlemcl  7831