Theorem caucvgprlemopl 7631

Description: Lemma for caucvgpr 7644. 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 5860	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2	1	breq1d 3999	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
3	2	rexbidv 2471	. . . . 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 5499	. . . . . 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 7325	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4133	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
8	6	rabex 4133	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
9	7, 8	op1st 6125	. . . . . 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 2191	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
11	3, 10	elrab2 2889	. . . 4 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
12	11	simprbi 273	. . 3 ⊢ (𝑠 ∈ (1st ‘𝐿) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
13	12	adantl 275	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
14		simprr 527	. . . 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 7377	. . . 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 121	. . 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 530	. . . . . . . . 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 7384	. . . . . . . . 9 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
19		recclnq 7354	. . . . . . . . 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 272	. . . . . . . . 9 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
22	21	ad3antlr 490	. . . . . . . 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 7369	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑗, 1o⟩] ~Q ) +Q 𝑠))
24	20, 22, 23	syl2anc 409	. . . . . . 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 7343	. . . . . . . 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 409	. . . . . . 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 4015	. . . . . 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 534	. . . . . 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 7360	. . . . . . 7 ⊢ <Q Or Q
30		ltrelnq 7327	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
31	29, 30	sotri 5006	. . . . . 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 409	. . . . 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 526	. . . . . 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 7370	. . . . . 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 409	. . . . 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 146	. . . 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 485	. . . . . . . . . . 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 485	. . . . . . . . . . 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 7343	. . . . . . . . . . 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 409	. . . . . . . . . 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 485	. . . . . . . . . 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 4013	. . . . . . . . 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 109	. . . . . . . . 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 4017	. . . . . . . 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 525	. . . . . . . . 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 7362	. . . . . . . . 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 1233	. . . . . . . 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 166	. . . . . . 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 485	. . . . . . . . 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 535	. . . . . . . . . . 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 485	. . . . . . . . . 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 7343	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 2205	. . . . . . . . . . 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 3999	. . . . . . . . . 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 166	. . . . . . . . 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 2519	. . . . . . . . 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 409	. . . . . . . 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 5860	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
60	59	breq1d 3999	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
61	60	rexbidv 2471	. . . . . . . . 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 2889	. . . . . . . 8 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
63	45, 58, 62	sylanbrc 415	. . . . . . 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 304	. . . . . 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 114	. . . . 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 2572	. . . 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 2592	. 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 2592	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘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  1_oc1o 6388  [cec 6511  Ncnpi 7234   <_N clti 7237   ~_Q ceq 7241  Qcnq 7242   +_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:  caucvgprlemrnd  7635