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Theorem caucvgprlemladdrl 7640

Description: Lemma for caucvgpr 7644. 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, 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}⟩
caucvgprlemladd.s	⊢ (𝜑 → 𝑆 ∈ Q)

**Assertion**
Ref	Expression
caucvgprlemladdrl	⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆)} ⊆ (1^st ‘(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 3765	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1_o⟩ = ⟨𝑎, 1_o⟩)
2	1	eceq1d 6549	. . . . . . . 8 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1_o⟩] ~_Q = [⟨𝑎, 1_o⟩] ~_Q )
3	2	fveq2d 5500	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) = (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))
4	3	oveq2d 5869	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) = (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )))
5		fveq2 5496	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
6	5	oveq1d 5868	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +_Q 𝑆) = ((𝐹‘𝑎) +_Q 𝑆))
7	4, 6	breq12d 4002	. . . . 5 ⊢ (𝑗 = 𝑎 → ((𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆) ↔ (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
8	7	cbvrexv 2697	. . . 4 ⊢ (∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆) ↔ ∃𝑎 ∈ N (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆))
9	8	a1i 9	. . 3 ⊢ (𝑙 ∈ Q → (∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆) ↔ ∃𝑎 ∈ N (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
10	9	rabbiia 2715	. 2 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆)} = {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)}
11		oveq1 5860	. . . . . . 7 ⊢ (𝑙 = 𝑟 → (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) = (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )))
12	11	breq1d 3999	. . . . . 6 ⊢ (𝑙 = 𝑟 → ((𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆) ↔ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
13	12	rexbidv 2471	. . . . 5 ⊢ (𝑙 = 𝑟 → (∃𝑎 ∈ N (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆) ↔ ∃𝑎 ∈ N (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
14	13	elrab 2886	. . . 4 ⊢ (𝑟 ∈ {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)} ↔ (𝑟 ∈ Q ∧ ∃𝑎 ∈ N (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
15		caucvgpr.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:N⟶Q)
16	15	ad4antr 491	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_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, 1_o⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )))))
18	17	ad4antr 491	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ∀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 )))))
19		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑏 ∈ N)
20		simpllr 529	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑎 ∈ N)
21	16, 18, 19, 20	caucvgprlemnbj 7629	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ¬ (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑎))
22	15	ad3antrrr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝐹:N⟶Q)
23	22	ffvelrnda 5631	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ Q)
24		nnnq 7384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ N → [⟨𝑏, 1_o⟩] ~_Q ∈ Q)
25		recclnq 7354	. . . . . . . . . . . . . . . . . 18 ⊢ ([⟨𝑏, 1_o⟩] ~_Q ∈ Q → (*_Q‘[⟨𝑏, 1_o⟩] ~_Q ) ∈ Q)
26	19, 24, 25	3syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) ∈ Q)
27		addclnq 7337	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑏) ∈ Q ∧ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) ∈ Q) → ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) ∈ Q)
28	23, 26, 27	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) ∈ Q)
29		nnnq 7384	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ N → [⟨𝑎, 1_o⟩] ~_Q ∈ Q)
30		recclnq 7354	. . . . . . . . . . . . . . . . 17 ⊢ ([⟨𝑎, 1_o⟩] ~_Q ∈ Q → (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q)
31	20, 29, 30	3syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q)
32		caucvgprlemladd.s	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ Q)
33	32	ad4antr 491	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑆 ∈ Q)
34		addassnqg 7344	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) ∈ Q ∧ (_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q ∧ 𝑆 ∈ Q) → ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) +_Q 𝑆) = (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q ((_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆)))
35	28, 31, 33, 34	syl3anc 1233	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) +_Q 𝑆) = (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q ((*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆)))
36	35	breq1d 3999	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) +_Q 𝑆) <_Q ((𝐹‘𝑎) +_Q 𝑆) ↔ (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q ((*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆)) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
37		ltanqg 7362	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
38	37	adantl 275	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_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 7337	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) ∈ Q ∧ (_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q) → (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ Q)
40	28, 31, 39	syl2anc 409	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ Q)
41	16, 20	ffvelrnd 5632	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (𝐹‘𝑎) ∈ Q)
42		addcomnqg 7343	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
43	42	adantl 275	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_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 6022	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑎) ↔ ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) +_Q 𝑆) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
45		addcomnqg 7343	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Q ∧ (_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q) → (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) = ((*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆))
46	33, 31, 45	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) = ((*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆))
47	46	oveq2d 5869	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) = (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q ((*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆)))
48	47	breq1d 3999	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆) ↔ (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q ((*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) +_Q 𝑆)) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
49	36, 44, 48	3bitr4rd 220	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆) ↔ (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑎)))
50	21, 49	mtbird 668	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → ¬ (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆))
51	50	nrexdv 2563	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ¬ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆))
52	51	intnand 926	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ¬ (((𝐹‘𝑎) +_Q 𝑆) ∈ Q ∧ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
53	17	ad3antrrr 489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ∀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 )))))
54		caucvgpr.bnd	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗))
55		fveq2 5496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑏 → (𝐹‘𝑗) = (𝐹‘𝑏))
56	55	breq2d 4001	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑏 → (𝐴 <_Q (𝐹‘𝑗) ↔ 𝐴 <_Q (𝐹‘𝑏)))
57	56	cbvralv 2696	. . . . . . . . . . . . . . 15 ⊢ (∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗) ↔ ∀𝑏 ∈ N 𝐴 <_Q (𝐹‘𝑏))
58	54, 57	sylib 121	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑏 ∈ N 𝐴 <_Q (𝐹‘𝑏))
59	58	ad3antrrr 489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_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, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q 𝑢}⟩
61		opeq1 3765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑏 → ⟨𝑗, 1_o⟩ = ⟨𝑏, 1_o⟩)
62	61	eceq1d 6549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑏 → [⟨𝑗, 1_o⟩] ~_Q = [⟨𝑏, 1_o⟩] ~_Q )
63	62	fveq2d 5500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑏 → (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) = (_Q‘[⟨𝑏, 1_o⟩] ~_Q ))
64	63	oveq2d 5869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑏 → (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) = (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )))
65	64, 55	breq12d 4002	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑏 → ((𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑏)))
66	65	cbvrexv 2697	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ ∃𝑏 ∈ N (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑏))
67	66	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ Q → (∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑗) ↔ ∃𝑏 ∈ N (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑏)))
68	67	rabbiia 2715	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑗)} = {𝑙 ∈ Q ∣ ∃𝑏 ∈ N (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑏)}
69	55, 63	oveq12d 5871	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑏 → ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) = ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )))
70	69	breq1d 3999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑏 → (((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q 𝑢 ↔ ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q 𝑢))
71	70	cbvrexv 2697	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q 𝑢 ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q 𝑢)
72	71	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ Q → (∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q 𝑢 ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q 𝑢))
73	72	rabbiia 2715	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑏 ∈ N ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q 𝑢}
74	68, 73	opeq12i 3770	. . . . . . . . . . . . . 14 ⊢ ⟨{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 ∣ ∃𝑏 ∈ N (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑏)}, {𝑢 ∈ Q ∣ ∃𝑏 ∈ N ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q 𝑢}⟩
75	60, 74	eqtri 2191	. . . . . . . . . . . . 13 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑏 ∈ N (𝑙 +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q (𝐹‘𝑏)}, {𝑢 ∈ Q ∣ ∃𝑏 ∈ N ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) <_Q 𝑢}⟩
76	32	ad3antrrr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝑆 ∈ Q)
77	29, 30	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ N → (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q)
78	77	ad2antlr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q)
79		addclnq 7337	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Q ∧ (_Q‘[⟨𝑎, 1_o⟩] ~_Q ) ∈ Q) → (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ Q)
80	76, 78, 79	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ Q)
81	22, 53, 59, 75, 80	caucvgprlemladdfu 7639	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) ⊆ {𝑢 ∈ Q ∣ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q 𝑢})
82	81	sseld 3146	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (((𝐹‘𝑎) +_Q 𝑆) ∈ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) → ((𝐹‘𝑎) +_Q 𝑆) ∈ {𝑢 ∈ Q ∣ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q 𝑢}))
83		breq2 3993	. . . . . . . . . . . . 13 ⊢ (𝑢 = ((𝐹‘𝑎) +_Q 𝑆) → ((((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q 𝑢 ↔ (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
84	83	rexbidv 2471	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝐹‘𝑎) +_Q 𝑆) → (∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q 𝑢 ↔ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
85	84	elrab 2886	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑎) +_Q 𝑆) ∈ {𝑢 ∈ Q ∣ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q 𝑢} ↔ (((𝐹‘𝑎) +_Q 𝑆) ∈ Q ∧ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆)))
86	82, 85	syl6ib 160	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (((𝐹‘𝑎) +_Q 𝑆) ∈ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) → (((𝐹‘𝑎) +_Q 𝑆) ∈ Q ∧ ∃𝑏 ∈ N (((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )) +_Q (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q ))) <_Q ((𝐹‘𝑎) +_Q 𝑆))))
87	52, 86	mtod 658	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ¬ ((𝐹‘𝑎) +_Q 𝑆) ∈ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)))
88	15, 17, 54, 60	caucvgprlemcl 7638	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ P)
89	88	ad3antrrr 489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝐿 ∈ P)
90		nqprlu 7509	. . . . . . . . . . . 12 ⊢ ((𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩ ∈ P)
91	80, 90	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩ ∈ P)
92		addclpr 7499	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩ ∈ P) → (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩) ∈ P)
93	89, 91, 92	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩) ∈ P)
94		prop 7437	. . . . . . . . . . 11 ⊢ ((𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩) ∈ P → ⟨(1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)), (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩))⟩ ∈ P)
95		prloc 7453	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)), (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩))⟩ ∈ P ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ((𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) ∨ ((𝐹‘𝑎) +_Q 𝑆) ∈ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩))))
96	94, 95	sylan 281	. . . . . . . . . 10 ⊢ (((𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩) ∈ P ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ((𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) ∨ ((𝐹‘𝑎) +_Q 𝑆) ∈ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩))))
97	93, 96	sylancom 418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ((𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) ∨ ((𝐹‘𝑎) +_Q 𝑆) ∈ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩))))
98	87, 97	ecased 1344	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)))
99		simpllr 529	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝑟 ∈ Q)
100	89, 76, 99, 78	caucvgprlemcanl 7606	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → ((𝑟 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q 𝑢}⟩)) ↔ 𝑟 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩))))
101	98, 100	mpbid 146	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝑟 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)))
102	101	ex 114	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) → ((𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆) → 𝑟 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩))))
103	102	rexlimdva 2587	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (∃𝑎 ∈ N (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆) → 𝑟 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩))))
104	103	expimpd 361	. . . 4 ⊢ (𝜑 → ((𝑟 ∈ Q ∧ ∃𝑎 ∈ N (𝑟 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝑟 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩))))
105	14, 104	syl5bi 151	. . 3 ⊢ (𝜑 → (𝑟 ∈ {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)} → 𝑟 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩))))
106	105	ssrdv 3153	. 2 ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑎 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑎, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)} ⊆ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)))
107	10, 106	eqsstrid 3193	1 ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑗, 1_o⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆)} ⊆ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∃wrex 2449 {crab 2452 ⊆ wss 3121 ⟨cop 3586 class class class wbr 3989 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 1^st c1st 6117 2^nd c2nd 6118 1_oc1o 6388 [cec 6511 Ncnpi 7234 <_N clti 7237 ~_Q ceq 7241 Qcnq 7242 +_Q cplq 7244 *_Qcrq 7246 <_Q cltq 7247 Pcnp 7253 +_P cpp 7255

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-2o 6396 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 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-iplp 7430 df-iltp 7432

This theorem is referenced by: caucvgprlem1 7641

W3C validator