Theorem caucvgprlemopl 7932

Description: Lemma for caucvgpr 7945. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-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
caucvgprlemopl	⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

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

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

Proof of Theorem caucvgprlemopl

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6035	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2	1	breq1d 4103	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
3	2	rexbidv 2534	. . . . 5 ⊢ (𝑙 = 𝑠 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
4		caucvgpr.lim	. . . . . . 7 ⊢ 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}⟩
5	4	fveq2i 5651	. . . . . 6 ⊢ (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}⟩)
6		nqex 7626	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4239	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
8	6	rabex 4239	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
9	7, 8	op1st 6318	. . . . . 6 ⊢ (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)}
10	5, 9	eqtri 2252	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
11	3, 10	elrab2 2966	. . . 4 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
12	11	simprbi 275	. . 3 ⊢ (𝑠 ∈ (1st ‘𝐿) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
13	12	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
14		simprr 533	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
15		ltbtwnnqq 7678	. . . 4 ⊢ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑡 ∈ Q ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))
16	14, 15	sylib 122	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) → ∃𝑡 ∈ Q ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))
17		simplrl 537	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → 𝑗 ∈ N)
18		nnnq 7685	. . . . . . . . 9 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
19		recclnq 7655	. . . . . . . . 9 ⊢ ([⟨𝑗, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
20	17, 18, 19	3syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
21	11	simplbi 274	. . . . . . . . 9 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
22	21	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → 𝑠 ∈ Q)
23		ltaddnq 7670	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
24	20, 22, 23	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
25		addcomnqg 7644	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
26	20, 22, 25	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
27	24, 26	breqtrd 4119	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
28		simprrl 541	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡)
29		ltsonq 7661	. . . . . . 7 ⊢ <Q Or Q
30		ltrelnq 7628	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
31	29, 30	sotri 5139	. . . . . 6 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑡)
32	27, 28, 31	syl2anc 411	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑡)
33		simprl 531	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → 𝑡 ∈ Q)
34		ltexnqq 7671	. . . . . 6 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝑡 ∈ Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟 ∈ Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡))
35	20, 33, 34	syl2anc 411	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑡 ↔ ∃𝑟 ∈ Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡))
36	32, 35	mpbid 147	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → ∃𝑟 ∈ Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡)
37	22	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑠 ∈ Q)
38	20	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
39		addcomnqg 7644	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
40	37, 38, 39	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
41	28	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡)
42	40, 41	eqbrtrrd 4117	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) <Q 𝑡)
43		simpr 110	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡)
44	42, 43	breqtrrd 4121	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟))
45		simplr 529	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑟 ∈ Q)
46		ltanqg 7663	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑟 ∈ Q ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q) → (𝑠 <Q 𝑟 ↔ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟)))
47	37, 45, 38, 46	syl3anc 1274	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ↔ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟)))
48	44, 47	mpbird 167	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑠 <Q 𝑟)
49	17	ad2antrr 488	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑗 ∈ N)
50		simprrr 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → 𝑡 <Q (𝐹‘𝑗))
51	50	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑡 <Q (𝐹‘𝑗))
52		addcomnqg 7644	. . . . . . . . . . . . 13 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝑟 ∈ Q) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
53	38, 45, 52	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
54	53, 43	eqtr3d 2266	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = 𝑡)
55	54	breq1d 4103	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → ((𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ 𝑡 <Q (𝐹‘𝑗)))
56	51, 55	mpbird 167	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
57		rspe 2582	. . . . . . . . 9 ⊢ ((𝑗 ∈ N ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
58	49, 56, 57	syl2anc 411	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
59		oveq1 6035	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
60	59	breq1d 4103	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
61	60	rexbidv 2534	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
62	61, 10	elrab2 2966	. . . . . . . 8 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
63	45, 58, 62	sylanbrc 417	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → 𝑟 ∈ (1st ‘𝐿))
64	48, 63	jca 306	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
65	64	ex 115	. . . . 5 ⊢ (((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) → (((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡 → (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))))
66	65	reximdva 2635	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → (∃𝑟 ∈ Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑟) = 𝑡 → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))))
67	36, 66	mpd 13	. . 3 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑗)))) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
68	16, 67	rexlimddv 2656	. 2 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
69	13, 68	rexlimddv 2656	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1^st c1st 6310  1_oc1o 6618  [cec 6743  Ncnpi 7535   <_N clti 7538   ~_Q ceq 7542  Qcnq 7543   +_Q cplq 7545  *_Qcrq 7547   <_Q cltq 7548

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616

This theorem is referenced by:  caucvgprlemrnd  7936