Theorem caucvgprlemopl 7753

Description: Lemma for caucvgpr 7766. 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 5932	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2	1	breq1d 4044	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
3	2	rexbidv 2498	. . . . 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 5564	. . . . . 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 7447	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4178	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
8	6	rabex 4178	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
9	7, 8	op1st 6213	. . . . . 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 2217	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
11	3, 10	elrab2 2923	. . . 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 531	. . . 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 7499	. . . 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 535	. . . . . . . . 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 7506	. . . . . . . . 9 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
19		recclnq 7476	. . . . . . . . 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 7491	. . . . . . . 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 7465	. . . . . . . 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 4060	. . . . . 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 539	. . . . . 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 7482	. . . . . . 7 ⊢ <Q Or Q
30		ltrelnq 7449	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
31	29, 30	sotri 5066	. . . . . 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 529	. . . . . 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 7492	. . . . . 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 7465	. . . . . . . . . . 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 4058	. . . . . . . . 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 4062	. . . . . . . 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 528	. . . . . . . . 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 7484	. . . . . . . . 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 1249	. . . . . . . 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 540	. . . . . . . . . . 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 7465	. . . . . . . . . . . . 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 2231	. . . . . . . . . . 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 4044	. . . . . . . . . 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 2546	. . . . . . . . 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 5932	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
60	59	breq1d 4044	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
61	60	rexbidv 2498	. . . . . . . . 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 2923	. . . . . . . 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 2599	. . . 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 2619	. 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 2619	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  ⟨cop 3626   class class class wbr 4034  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  1_oc1o 6476  [cec 6599  Ncnpi 7356   <_N clti 7359   ~_Q ceq 7363  Qcnq 7364   +_Q cplq 7366  *_Qcrq 7368   <_Q cltq 7369

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437

This theorem is referenced by:  caucvgprlemrnd  7757