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Theorem caucvgprlemopl 6859

Description: Lemma for caucvgpr 6872. 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, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩

**Assertion**
Ref	Expression
caucvgprlemopl	⊢ ((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘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 5539	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
2	1	breq1d 3795	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)))
3	2	rexbidv 2369	. . . . 5 ⊢ (𝑙 = 𝑠 → (∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)))
4		caucvgpr.lim	. . . . . . 7 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩
5	4	fveq2i 5201	. . . . . 6 ⊢ (1^st ‘𝐿) = (1^st ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩)
6		nqex 6553	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 3922	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)} ∈ V
8	6	rabex 3922	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢} ∈ V
9	7, 8	op1st 5793	. . . . . 6 ⊢ (1^st ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}
10	5, 9	eqtri 2101	. . . . 5 ⊢ (1^st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}
11	3, 10	elrab2 2751	. . . 4 ⊢ (𝑠 ∈ (1^st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)))
12	11	simprbi 269	. . 3 ⊢ (𝑠 ∈ (1^st ‘𝐿) → ∃𝑗 ∈ N (𝑠 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))
13	12	adantl 271	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) → ∃𝑗 ∈ N (𝑠 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))
14		simprr 498	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) → (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))
15		ltbtwnnqq 6605	. . . 4 ⊢ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ ∃𝑡 ∈ Q ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))
16	14, 15	sylib 120	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) → ∃𝑡 ∈ Q ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))
17		simplrl 501	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → 𝑗 ∈ N)
18		nnnq 6612	. . . . . . . . 9 ⊢ (𝑗 ∈ N → [⟨𝑗, 1_𝑜⟩] ~_Q ∈ Q)
19		recclnq 6582	. . . . . . . . 9 ⊢ ([⟨𝑗, 1_𝑜⟩] ~_Q ∈ Q → (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q)
20	17, 18, 19	3syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q)
21	11	simplbi 268	. . . . . . . . 9 ⊢ (𝑠 ∈ (1^st ‘𝐿) → 𝑠 ∈ Q)
22	21	ad3antlr 476	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → 𝑠 ∈ Q)
23		ltaddnq 6597	. . . . . . . 8 ⊢ (((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q ∧ 𝑠 ∈ Q) → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠))
24	20, 22, 23	syl2anc 403	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠))
25		addcomnqg 6571	. . . . . . . 8 ⊢ (((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q ∧ 𝑠 ∈ Q) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠) = (𝑠 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
26	20, 22, 25	syl2anc 403	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠) = (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
27	24, 26	breqtrd 3809	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
28		simprrl 505	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → (𝑠 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡)
29		ltsonq 6588	. . . . . . 7 ⊢ <_Q Or Q
30		ltrelnq 6555	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
31	29, 30	sotri 4740	. . . . . 6 ⊢ (((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡) → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑡)
32	27, 28, 31	syl2anc 403	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑡)
33		simprl 497	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → 𝑡 ∈ Q)
34		ltexnqq 6598	. . . . . 6 ⊢ (((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q ∧ 𝑡 ∈ Q) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑡 ↔ ∃𝑟 ∈ Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡))
35	20, 33, 34	syl2anc 403	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑡 ↔ ∃𝑟 ∈ Q ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡))
36	32, 35	mpbid 145	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → ∃𝑟 ∈ Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡)
37	22	ad2antrr 471	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → 𝑠 ∈ Q)
38	20	ad2antrr 471	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q)
39		addcomnqg 6571	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Q ∧ (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q) → (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠))
40	37, 38, 39	syl2anc 403	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠))
41	28	ad2antrr 471	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡)
42	40, 41	eqbrtrrd 3807	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠) <_Q 𝑡)
43		simpr 108	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡)
44	42, 43	breqtrrd 3811	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠) <_Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟))
45		simplr 496	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → 𝑟 ∈ Q)
46		ltanqg 6590	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑟 ∈ Q ∧ (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q) → (𝑠 <_Q 𝑟 ↔ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠) <_Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟)))
47	37, 45, 38, 46	syl3anc 1169	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (𝑠 <_Q 𝑟 ↔ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑠) <_Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟)))
48	44, 47	mpbird 165	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → 𝑠 <_Q 𝑟)
49	17	ad2antrr 471	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → 𝑗 ∈ N)
50		simprrr 506	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → 𝑡 <_Q (𝐹‘𝑗))
51	50	ad2antrr 471	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → 𝑡 <_Q (𝐹‘𝑗))
52		addcomnqg 6571	. . . . . . . . . . . . 13 ⊢ (((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q ∧ 𝑟 ∈ Q) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = (𝑟 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
53	38, 45, 52	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = (𝑟 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
54	53, 43	eqtr3d 2115	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = 𝑡)
55	54	breq1d 3795	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → ((𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ 𝑡 <_Q (𝐹‘𝑗)))
56	51, 55	mpbird 165	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))
57		rspe 2412	. . . . . . . . 9 ⊢ ((𝑗 ∈ N ∧ (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)) → ∃𝑗 ∈ N (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))
58	49, 56, 57	syl2anc 403	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → ∃𝑗 ∈ N (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))
59		oveq1 5539	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )))
60	59	breq1d 3795	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)))
61	60	rexbidv 2369	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑟 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)))
62	61, 10	elrab2 2751	. . . . . . . 8 ⊢ (𝑟 ∈ (1^st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N (𝑟 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)))
63	45, 58, 62	sylanbrc 408	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → 𝑟 ∈ (1^st ‘𝐿))
64	48, 63	jca 300	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) ∧ ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡) → (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)))
65	64	ex 113	. . . . 5 ⊢ (((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) ∧ 𝑟 ∈ Q) → (((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡 → (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿))))
66	65	reximdva 2463	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → (∃𝑟 ∈ Q ((*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) +_Q 𝑟) = 𝑡 → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿))))
67	36, 66	mpd 13	. . 3 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑡 ∧ 𝑡 <_Q (𝐹‘𝑗)))) → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)))
68	16, 67	rexlimddv 2481	. 2 ⊢ (((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) ∧ (𝑗 ∈ N ∧ (𝑠 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗))) → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)))
69	13, 68	rexlimddv 2481	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 {crab 2352 ⟨cop 3401 class class class wbr 3785 ⟶wf 4918 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785 1_𝑜c1o 6017 [cec 6127 Ncnpi 6462 <_N clti 6465 ~_Q ceq 6469 Qcnq 6470 +_Q cplq 6472 *_Qcrq 6474 <_Q cltq 6475

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543

This theorem is referenced by: caucvgprlemrnd 6863

W3C validator