Theorem caucvgprlemladdrl 7677

Description: Lemma for caucvgpr 7681. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-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}⟩
caucvgprlemladd.s	⊢ (𝜑 → 𝑆 ∈ Q)

Assertion

Ref	Expression
caucvgprlemladdrl	⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑢,𝑙   𝑛,𝐹,𝑘   𝑘,𝐿,𝑗   𝑆,𝑙,𝑢,𝑗   𝑗,𝑘,𝑆

Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝑆(𝑛)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlemladdrl

Dummy variables 𝑟 𝑓 𝑔 ℎ 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3779	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
2	1	eceq1d 6571	. . . . . . . 8 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
3	2	fveq2d 5520	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
4	3	oveq2d 5891	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
5		fveq2 5516	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
6	5	oveq1d 5890	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +Q 𝑆) = ((𝐹‘𝑎) +Q 𝑆))
7	4, 6	breq12d 4017	. . . . 5 ⊢ (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)))
8	7	cbvrexv 2705	. . . 4 ⊢ (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆) ↔ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆))
9	8	a1i 9	. . 3 ⊢ (𝑙 ∈ Q → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆) ↔ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)))
10	9	rabbiia 2723	. 2 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆)} = {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)}
11		oveq1 5882	. . . . . . 7 ⊢ (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
12	11	breq1d 4014	. . . . . 6 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆) ↔ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)))
13	12	rexbidv 2478	. . . . 5 ⊢ (𝑙 = 𝑟 → (∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆) ↔ ∃𝑎 ∈ N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)))
14	13	elrab 2894	. . . 4 ⊢ (𝑟 ∈ {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)} ↔ (𝑟 ∈ Q ∧ ∃𝑎 ∈ N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)))
15		caucvgpr.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:N⟶Q)
16	15	ad4antr 494	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → 𝐹:N⟶Q)
17		caucvgpr.cau	. . . . . . . . . . . . . . 15 ⊢ (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 )))))
18	17	ad4antr 494	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ∀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 )))))
19		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑏 ∈ N)
20		simpllr 534	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑎 ∈ N)
21	16, 18, 19, 20	caucvgprlemnbj 7666	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ¬ (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎))
22	15	ad3antrrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → 𝐹:N⟶Q)
23	22	ffvelcdmda 5652	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ Q)
24		nnnq 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ N → [⟨𝑏, 1o⟩] ~Q ∈ Q)
25		recclnq 7391	. . . . . . . . . . . . . . . . . 18 ⊢ ([⟨𝑏, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
26	19, 24, 25	3syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
27		addclnq 7374	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑏) ∈ Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
28	23, 26, 27	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
29		nnnq 7421	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ N → [⟨𝑎, 1o⟩] ~Q ∈ Q)
30		recclnq 7391	. . . . . . . . . . . . . . . . 17 ⊢ ([⟨𝑎, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
31	20, 29, 30	3syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
32		caucvgprlemladd.s	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ Q)
33	32	ad4antr 494	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑆 ∈ Q)
34		addassnqg 7381	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q ∧ 𝑆 ∈ Q) → ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)))
35	28, 31, 33, 34	syl3anc 1238	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)))
36	35	breq1d 4014	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) <Q ((𝐹‘𝑎) +Q 𝑆) ↔ (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)) <Q ((𝐹‘𝑎) +Q 𝑆)))
37		ltanqg 7399	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
38	37	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
39		addclnq 7374	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q) → (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
40	28, 31, 39	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
41	16, 20	ffvelcdmd 5653	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (𝐹‘𝑎) ∈ Q)
42		addcomnqg 7380	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
43	42	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
44	38, 40, 41, 33, 43	caovord2d 6044	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎) ↔ ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) <Q ((𝐹‘𝑎) +Q 𝑆)))
45		addcomnqg 7380	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆))
46	33, 31, 45	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆))
47	46	oveq2d 5891	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) = (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)))
48	47	breq1d 4014	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆) ↔ (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)) <Q ((𝐹‘𝑎) +Q 𝑆)))
49	36, 44, 48	3bitr4rd 221	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆) ↔ (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹‘𝑎)))
50	21, 49	mtbird 673	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) ∧ 𝑏 ∈ N) → ¬ (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆))
51	50	nrexdv 2570	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ¬ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆))
52	51	intnand 931	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ¬ (((𝐹‘𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆)))
53	17	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ∀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 )))))
54		caucvgpr.bnd	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
55		fveq2 5516	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑏 → (𝐹‘𝑗) = (𝐹‘𝑏))
56	55	breq2d 4016	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑏 → (𝐴 <Q (𝐹‘𝑗) ↔ 𝐴 <Q (𝐹‘𝑏)))
57	56	cbvralv 2704	. . . . . . . . . . . . . . 15 ⊢ (∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗) ↔ ∀𝑏 ∈ N 𝐴 <Q (𝐹‘𝑏))
58	54, 57	sylib 122	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑏 ∈ N 𝐴 <Q (𝐹‘𝑏))
59	58	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ∀𝑏 ∈ N 𝐴 <Q (𝐹‘𝑏))
60		caucvgpr.lim	. . . . . . . . . . . . . 14 ⊢ 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}⟩
61		opeq1 3779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑏 → ⟨𝑗, 1o⟩ = ⟨𝑏, 1o⟩)
62	61	eceq1d 6571	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑏 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
63	62	fveq2d 5520	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑏 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
64	63	oveq2d 5891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑏 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
65	64, 55	breq12d 4017	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑏 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹‘𝑏)))
66	65	cbvrexv 2705	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑏 ∈ N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹‘𝑏))
67	66	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ Q → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑏 ∈ N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹‘𝑏)))
68	67	rabbiia 2723	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} = {𝑙 ∈ Q ∣ ∃𝑏 ∈ N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹‘𝑏)}
69	55, 63	oveq12d 5893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑏 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
70	69	breq1d 4014	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑏 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢))
71	70	cbvrexv 2705	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢)
72	71	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ Q → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢))
73	72	rabbiia 2723	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑏 ∈ N ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢}
74	68, 73	opeq12i 3784	. . . . . . . . . . . . . 14 ⊢ ⟨{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 ∣ ∃𝑏 ∈ N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹‘𝑏)}, {𝑢 ∈ Q ∣ ∃𝑏 ∈ N ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢}⟩
75	60, 74	eqtri 2198	. . . . . . . . . . . . 13 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑏 ∈ N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹‘𝑏)}, {𝑢 ∈ Q ∣ ∃𝑏 ∈ N ((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢}⟩
76	32	ad3antrrr 492	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → 𝑆 ∈ Q)
77	29, 30	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ N → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
78	77	ad2antlr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
79		addclnq 7374	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
80	76, 78, 79	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
81	22, 53, 59, 75, 80	caucvgprlemladdfu 7676	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ⊆ {𝑢 ∈ Q ∣ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢})
82	81	sseld 3155	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (((𝐹‘𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) → ((𝐹‘𝑎) +Q 𝑆) ∈ {𝑢 ∈ Q ∣ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢}))
83		breq2 4008	. . . . . . . . . . . . 13 ⊢ (𝑢 = ((𝐹‘𝑎) +Q 𝑆) → ((((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢 ↔ (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆)))
84	83	rexbidv 2478	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝐹‘𝑎) +Q 𝑆) → (∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢 ↔ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆)))
85	84	elrab 2894	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑎) +Q 𝑆) ∈ {𝑢 ∈ Q ∣ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢} ↔ (((𝐹‘𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆)))
86	82, 85	imbitrdi 161	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (((𝐹‘𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) → (((𝐹‘𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏 ∈ N (((𝐹‘𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹‘𝑎) +Q 𝑆))))
87	52, 86	mtod 663	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ¬ ((𝐹‘𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)))
88	15, 17, 54, 60	caucvgprlemcl 7675	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ P)
89	88	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → 𝐿 ∈ P)
90		nqprlu 7546	. . . . . . . . . . . 12 ⊢ ((𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
91	80, 90	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
92		addclpr 7536	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P)
93	89, 91, 92	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P)
94		prop 7474	. . . . . . . . . . 11 ⊢ ((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)), (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))⟩ ∈ P)
95		prloc 7490	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)), (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))⟩ ∈ P ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹‘𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))))
96	94, 95	sylan 283	. . . . . . . . . 10 ⊢ (((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹‘𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))))
97	93, 96	sylancom 420	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹‘𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))))
98	87, 97	ecased 1349	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)))
99		simpllr 534	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → 𝑟 ∈ Q)
100	89, 76, 99, 78	caucvgprlemcanl 7643	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ↔ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
101	98, 100	mpbid 147	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
102	101	ex 115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
103	102	rexlimdva 2594	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (∃𝑎 ∈ N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
104	103	expimpd 363	. . . 4 ⊢ (𝜑 → ((𝑟 ∈ Q ∧ ∃𝑎 ∈ N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
105	14, 104	biimtrid 152	. . 3 ⊢ (𝜑 → (𝑟 ∈ {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)} → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
106	105	ssrdv 3162	. 2 ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹‘𝑎) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
107	10, 106	eqsstrid 3202	1 ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3130  ⟨cop 3596   class class class wbr 4004  ⟶wf 5213  ‘cfv 5217  (class class class)co 5875  1^st c1st 6139  2^nd c2nd 6140  1_oc1o 6410  [cec 6533  Ncnpi 7271   <_N clti 7274   ~_Q ceq 7278  Qcnq 7279   +_Q cplq 7281  *_Qcrq 7283   <_Q cltq 7284  Pcnp 7290   +_P cpp 7292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469

This theorem is referenced by:  caucvgprlem1  7678