Theorem caucvgprlemm 7979

Description: Lemma for caucvgpr 7993. 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 5669	. . . . . 6 ⊢ (𝑗 = 1o → (𝐹‘𝑗) = (𝐹‘1o))
2	1	breq2d 4120	. . . . 5 ⊢ (𝑗 = 1o → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘1o)))
3		caucvgpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
4		1pi 7626	. . . . . 6 ⊢ 1o ∈ N
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → 1o ∈ N)
6	2, 3, 5	rspcdva 2925	. . . 4 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1o))
7		ltrelnq 7676	. . . . . 6 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4801	. . . . 5 ⊢ (𝐴 <Q (𝐹‘1o) → (𝐴 ∈ Q ∧ (𝐹‘1o) ∈ Q))
9	8	simpld 112	. . . 4 ⊢ (𝐴 <Q (𝐹‘1o) → 𝐴 ∈ Q)
10		halfnqq 7721	. . . 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 529	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
13		archrecnq 7974	. . . . . . . 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 529	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → 𝑗 ∈ N)
17		nnnq 7733	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
18		recclnq 7703	. . . . . . . . . . . . . 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 7711	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 536	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) = 𝐴)
25	23, 24	breqtrd 4134	. . . . . . . . . 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 2589	. . . . . . . . . . . . 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 7709	. . . . . . . . . . 11 ⊢ <Q Or Q
31	30, 7	sotri 5157	. . . . . . . . . 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 2644	. . . . . . 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 6056	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
37	36	breq1d 4118	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
38	37	rexbidv 2543	. . . . . . 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 5672	. . . . . . . 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 7674	. . . . . . . . . 10 ⊢ Q ∈ V
42	41	rabex 4255	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
43	41	rabex 4255	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
44	42, 43	op1st 6339	. . . . . . . 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 2253	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
46	38, 45	elrab2 2975	. . . . . 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 2644	. . 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 5812	. . . . 5 ⊢ (𝜑 → (𝐹‘1o) ∈ Q)
53		1nq 7677	. . . . 5 ⊢ 1Q ∈ Q
54		addclnq 7686	. . . . 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 7686	. . . 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 7662	. . . . . . . . 9 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
59	58	fveq2i 5672	. . . . . . . 8 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
60		rec1nq 7706	. . . . . . . 8 ⊢ (*Q‘1Q) = 1Q
61	59, 60	eqtr3i 2255	. . . . . . 7 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
62	61	oveq2i 6060	. . . . . 6 ⊢ ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) = ((𝐹‘1o) +Q 1Q)
63		ltaddnq 7718	. . . . . . 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 4143	. . . . 5 ⊢ (𝜑 → ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )) <Q (((𝐹‘1o) +Q 1Q) +Q 1Q))
66		opeq1 3882	. . . . . . . . . 10 ⊢ (𝑗 = 1o → ⟨𝑗, 1o⟩ = ⟨1o, 1o⟩)
67	66	eceq1d 6802	. . . . . . . . 9 ⊢ (𝑗 = 1o → [⟨𝑗, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
68	67	fveq2d 5673	. . . . . . . 8 ⊢ (𝑗 = 1o → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
69	1, 68	oveq12d 6067	. . . . . . 7 ⊢ (𝑗 = 1o → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘1o) +Q (*Q‘[⟨1o, 1o⟩] ~Q )))
70	69	breq1d 4118	. . . . . 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 2920	. . . . 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 4112	. . . . . 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 2543	. . . . 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 5672	. . . . . 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 6340	. . . . . 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 2253	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
78	74, 77	elrab2 2975	. . . 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 2295	. . . 4 ⊢ (𝑟 = (((𝐹‘1o) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1o) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
81	80	rspcev 2920	. . 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 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524  ⟨cop 3691   class class class wbr 4108  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  1^st c1st 6331  2^nd c2nd 6332  1_oc1o 6639  [cec 6764  Ncnpi 7583   <_N clti 7586   ~_Q ceq 7590  Qcnq 7591  1_Qc1q 7592   +_Q cplq 7593  *_Qcrq 7595   <_Q cltq 7596

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664

This theorem is referenced by:  caucvgprlemcl  7987