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Theorem caucvgprlemladdrl 6776

Description: Lemma for caucvgpr 6780. 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 3549	. . . . . . . . 9 ⊢ (𝑗 = 𝑎 → ⟨𝑗, 1_𝑜⟩ = ⟨𝑎, 1_𝑜⟩)
2	1	eceq1d 6142	. . . . . . . 8 ⊢ (𝑗 = 𝑎 → [⟨𝑗, 1_𝑜⟩] ~_Q = [⟨𝑎, 1_𝑜⟩] ~_Q )
3	2	fveq2d 5182	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) = (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q ))
4	3	oveq2d 5528	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = (𝑙 +_Q (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )))
5		fveq2 5178	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
6	5	oveq1d 5527	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑗) +_Q 𝑆) = ((𝐹‘𝑎) +_Q 𝑆))
7	4, 6	breq12d 3777	. . . . 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 2534	. . . 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 2547	. 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 5519	. . . . . . 7 ⊢ (𝑙 = 𝑟 → (𝑙 +_Q (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )) = (𝑟 +_Q (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )))
12	11	breq1d 3774	. . . . . 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 2327	. . . . 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 2698	. . . 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 463	. . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . 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 103	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑏 ∈ N)
20		simpllr 486	. . . . . . . . . . . . . 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 6765	. . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝐹:N⟶Q)
23	22	ffvelrnda 5302	. . . . . . . . . . . . . . . . 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 6520	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ N → [⟨𝑏, 1_𝑜⟩] ~_Q ∈ Q)
25		recclnq 6490	. . . . . . . . . . . . . . . . . 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 6473	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑏) ∈ Q ∧ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) ∈ Q) → ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )) ∈ Q)
28	23, 26, 27	syl2anc 391	. . . . . . . . . . . . . . . 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 6520	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ N → [⟨𝑎, 1_𝑜⟩] ~_Q ∈ Q)
30		recclnq 6490	. . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) ∧ 𝑏 ∈ N) → 𝑆 ∈ Q)
34		addassnqg 6480	. . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . 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 3774	. . . . . . . . . . . . . 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 6498	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
38	37	adantl 262	. . . . . . . . . . . . . . 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 6473	. . . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . . 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 5303	. . . . . . . . . . . . . . 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 6479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
43	42	adantl 262	. . . . . . . . . . . . . . 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 5670	. . . . . . . . . . . . . 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 6479	. . . . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . . . 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 5528	. . . . . . . . . . . . . . 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 3774	. . . . . . . . . . . . . 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 210	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 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 2412	. . . . . . . . . . 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 840	. . . . . . . . . 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 461	. . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑏 → (𝐹‘𝑗) = (𝐹‘𝑏))
56	55	breq2d 3776	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑏 → (𝐴 <_Q (𝐹‘𝑗) ↔ 𝐴 <_Q (𝐹‘𝑏)))
57	56	cbvralv 2533	. . . . . . . . . . . . . . 15 ⊢ (∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗) ↔ ∀𝑏 ∈ N 𝐴 <_Q (𝐹‘𝑏))
58	54, 57	sylib 127	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑏 ∈ N 𝐴 <_Q (𝐹‘𝑏))
59	58	ad3antrrr 461	. . . . . . . . . . . . 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 3549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑏 → ⟨𝑗, 1_𝑜⟩ = ⟨𝑏, 1_𝑜⟩)
62	61	eceq1d 6142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑏 → [⟨𝑗, 1_𝑜⟩] ~_Q = [⟨𝑏, 1_𝑜⟩] ~_Q )
63	62	fveq2d 5182	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑏 → (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) = (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ))
64	63	oveq2d 5528	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑏 → (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = (𝑙 +_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )))
65	64, 55	breq12d 3777	. . . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 2547	. . . . . . . . . . . . . . 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 5530	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑏 → ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) = ((𝐹‘𝑏) +_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )))
70	69	breq1d 3774	. . . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 2547	. . . . . . . . . . . . . . 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 3554	. . . . . . . . . . . . . 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 2060	. . . . . . . . . . . . 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 461	. . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . 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 6473	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Q ∧ (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q ) ∈ Q) → (𝑆 +_Q (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )) ∈ Q)
80	76, 78, 79	syl2anc 391	. . . . . . . . . . . . 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 6775	. . . . . . . . . . . 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 2944	. . . . . . . . . . 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 3768	. . . . . . . . . . . . 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 2327	. . . . . . . . . . . 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 2698	. . . . . . . . . . 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 150	. . . . . . . . . 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 589	. . . . . . . . 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 6774	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ P)
89	88	ad3antrrr 461	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑎 ∈ N) ∧ (𝑟 +_Q (*_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )) <_Q ((𝐹‘𝑎) +_Q 𝑆)) → 𝐿 ∈ P)
90		nqprlu 6645	. . . . . . . . . . . 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 6635	. . . . . . . . . . 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 391	. . . . . . . . . 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 6573	. . . . . . . . . . 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 6589	. . . . . . . . . . 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 267	. . . . . . . . . 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 397	. . . . . . . . 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 1239	. . . . . . . 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 486	. . . . . . . . 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 6742	. . . . . . . 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 135	. . . . . . 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 108	. . . . . 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 2433	. . . . 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 345	. . . 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 141	. . 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 2951	. 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	syl5eqss 2989	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 97 ↔ wb 98 ∨ wo 629 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∃wrex 2307 {crab 2310 ⊆ wss 2917 ⟨cop 3378 class class class wbr 3764 ⟶wf 4898 ‘cfv 4902 (class class class)co 5512 1^st c1st 5765 2^nd c2nd 5766 1_𝑜c1o 5994 [cec 6104 Ncnpi 6370 <_N clti 6373 ~_Q ceq 6377 Qcnq 6378 +_Q cplq 6380 *_Qcrq 6382 <_Q cltq 6383 Pcnp 6389 +_P cpp 6391

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-iplp 6566 df-iltp 6568

This theorem is referenced by: caucvgprlem1 6777

W3C validator