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Theorem caucvgprlemladdrl 7676
Description: Lemma for caucvgpr 7680. 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 (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
caucvgprlemladd.s (𝜑𝑆Q)
Assertion
Ref Expression
caucvgprlemladdrl (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <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 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3778 . . . . . . . . 9 (𝑗 = 𝑎 → ⟨𝑗, 1o⟩ = ⟨𝑎, 1o⟩)
21eceq1d 6570 . . . . . . . 8 (𝑗 = 𝑎 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
32fveq2d 5519 . . . . . . 7 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
43oveq2d 5890 . . . . . 6 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
5 fveq2 5515 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
65oveq1d 5889 . . . . . 6 (𝑗 = 𝑎 → ((𝐹𝑗) +Q 𝑆) = ((𝐹𝑎) +Q 𝑆))
74, 6breq12d 4016 . . . . 5 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
87cbvrexv 2704 . . . 4 (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆))
98a1i 9 . . 3 (𝑙Q → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
109rabbiia 2722 . 2 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} = {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)}
11 oveq1 5881 . . . . . . 7 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
1211breq1d 4013 . . . . . 6 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) ↔ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
1312rexbidv 2478 . . . . 5 (𝑙 = 𝑟 → (∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) ↔ ∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
1413elrab 2893 . . . 4 (𝑟 ∈ {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)} ↔ (𝑟Q ∧ ∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
15 caucvgpr.f . . . . . . . . . . . . . . 15 (𝜑𝐹:NQ)
1615ad4antr 494 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝐹:NQ)
17 caucvgpr.cau . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
1817ad4antr 494 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
19 simpr 110 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝑏N)
20 simpllr 534 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝑎N)
2116, 18, 19, 20caucvgprlemnbj 7665 . . . . . . . . . . . . 13 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ¬ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎))
2215ad3antrrr 492 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝐹:NQ)
2322ffvelcdmda 5651 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (𝐹𝑏) ∈ Q)
24 nnnq 7420 . . . . . . . . . . . . . . . . . 18 (𝑏N → [⟨𝑏, 1o⟩] ~QQ)
25 recclnq 7390 . . . . . . . . . . . . . . . . . 18 ([⟨𝑏, 1o⟩] ~QQ → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
2619, 24, 253syl 17 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
27 addclnq 7373 . . . . . . . . . . . . . . . . 17 (((𝐹𝑏) ∈ Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
2823, 26, 27syl2anc 411 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
29 nnnq 7420 . . . . . . . . . . . . . . . . 17 (𝑎N → [⟨𝑎, 1o⟩] ~QQ)
30 recclnq 7390 . . . . . . . . . . . . . . . . 17 ([⟨𝑎, 1o⟩] ~QQ → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
3120, 29, 303syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
32 caucvgprlemladd.s . . . . . . . . . . . . . . . . 17 (𝜑𝑆Q)
3332ad4antr 494 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝑆Q)
34 addassnqg 7380 . . . . . . . . . . . . . . . 16 ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q𝑆Q) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)))
3528, 31, 33, 34syl3anc 1238 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)))
3635breq1d 4013 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) <Q ((𝐹𝑎) +Q 𝑆) ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)) <Q ((𝐹𝑎) +Q 𝑆)))
37 ltanqg 7398 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3837adantl 277 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
39 addclnq 7373 . . . . . . . . . . . . . . . 16 ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q) → (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
4028, 31, 39syl2anc 411 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
4116, 20ffvelcdmd 5652 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (𝐹𝑎) ∈ Q)
42 addcomnqg 7379 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4342adantl 277 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4438, 40, 41, 33, 43caovord2d 6043 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎) ↔ ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) +Q 𝑆) <Q ((𝐹𝑎) +Q 𝑆)))
45 addcomnqg 7379 . . . . . . . . . . . . . . . . 17 ((𝑆Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆))
4633, 31, 45syl2anc 411 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆))
4746oveq2d 5890 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) = (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)))
4847breq1d 4013 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆) ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1o⟩] ~Q ) +Q 𝑆)) <Q ((𝐹𝑎) +Q 𝑆)))
4936, 44, 483bitr4rd 221 . . . . . . . . . . . . 13 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆) ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q (𝐹𝑎)))
5021, 49mtbird 673 . . . . . . . . . . . 12 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ¬ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆))
5150nrexdv 2570 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ¬ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆))
5251intnand 931 . . . . . . . . . 10 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ¬ (((𝐹𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
5317ad3antrrr 492 . . . . . . . . . . . . 13 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
54 caucvgpr.bnd . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
55 fveq2 5515 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑏 → (𝐹𝑗) = (𝐹𝑏))
5655breq2d 4015 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑏 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹𝑏)))
5756cbvralv 2703 . . . . . . . . . . . . . . 15 (∀𝑗N 𝐴 <Q (𝐹𝑗) ↔ ∀𝑏N 𝐴 <Q (𝐹𝑏))
5854, 57sylib 122 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑏N 𝐴 <Q (𝐹𝑏))
5958ad3antrrr 492 . . . . . . . . . . . . 13 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ∀𝑏N 𝐴 <Q (𝐹𝑏))
60 caucvgpr.lim . . . . . . . . . . . . . 14 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
61 opeq1 3778 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑏 → ⟨𝑗, 1o⟩ = ⟨𝑏, 1o⟩)
6261eceq1d 6570 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑏 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
6362fveq2d 5519 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑏 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
6463oveq2d 5890 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑏 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
6564, 55breq12d 4016 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑏 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹𝑏)))
6665cbvrexv 2704 . . . . . . . . . . . . . . . . 17 (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹𝑏))
6766a1i 9 . . . . . . . . . . . . . . . 16 (𝑙Q → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹𝑏)))
6867rabbiia 2722 . . . . . . . . . . . . . . 15 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙Q ∣ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹𝑏)}
6955, 63oveq12d 5892 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑏 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
7069breq1d 4013 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑏 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢))
7170cbvrexv 2704 . . . . . . . . . . . . . . . . 17 (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢)
7271a1i 9 . . . . . . . . . . . . . . . 16 (𝑢Q → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢))
7372rabbiia 2722 . . . . . . . . . . . . . . 15 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢}
7468, 73opeq12i 3783 . . . . . . . . . . . . . 14 ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙Q ∣ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹𝑏)}, {𝑢Q ∣ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢}⟩
7560, 74eqtri 2198 . . . . . . . . . . . . 13 𝐿 = ⟨{𝑙Q ∣ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝐹𝑏)}, {𝑢Q ∣ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑢}⟩
7632ad3antrrr 492 . . . . . . . . . . . . . 14 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑆Q)
7729, 30syl 14 . . . . . . . . . . . . . . 15 (𝑎N → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
7877ad2antlr 489 . . . . . . . . . . . . . 14 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q)
79 addclnq 7373 . . . . . . . . . . . . . 14 ((𝑆Q ∧ (*Q‘[⟨𝑎, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
8076, 78, 79syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q)
8122, 53, 59, 75, 80caucvgprlemladdfu 7675 . . . . . . . . . . . 12 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ⊆ {𝑢Q ∣ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢})
8281sseld 3154 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) → ((𝐹𝑎) +Q 𝑆) ∈ {𝑢Q ∣ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢}))
83 breq2 4007 . . . . . . . . . . . . 13 (𝑢 = ((𝐹𝑎) +Q 𝑆) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢 ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
8483rexbidv 2478 . . . . . . . . . . . 12 (𝑢 = ((𝐹𝑎) +Q 𝑆) → (∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢 ↔ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
8584elrab 2893 . . . . . . . . . . 11 (((𝐹𝑎) +Q 𝑆) ∈ {𝑢Q ∣ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q 𝑢} ↔ (((𝐹𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
8682, 85imbitrdi 161 . . . . . . . . . 10 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) → (((𝐹𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆))))
8752, 86mtod 663 . . . . . . . . 9 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ¬ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)))
8815, 17, 54, 60caucvgprlemcl 7674 . . . . . . . . . . . 12 (𝜑𝐿P)
8988ad3antrrr 492 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝐿P)
90 nqprlu 7545 . . . . . . . . . . . 12 ((𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ Q → ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
9180, 90syl 14 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
92 addclpr 7535 . . . . . . . . . . 11 ((𝐿P ∧ ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P)
9389, 91, 92syl2anc 411 . . . . . . . . . 10 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P)
94 prop 7473 . . . . . . . . . . 11 ((𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)), (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))⟩ ∈ P)
95 prloc 7489 . . . . . . . . . . 11 ((⟨(1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)), (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))⟩ ∈ P ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))))
9694, 95sylan 283 . . . . . . . . . 10 (((𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩) ∈ P ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))))
9793, 96sylancom 420 . . . . . . . . 9 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩))))
9887, 97ecased 1349 . . . . . . . 8 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)))
99 simpllr 534 . . . . . . . . 9 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑟Q)
10089, 76, 99, 78caucvgprlemcanl 7642 . . . . . . . 8 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑢}⟩)) ↔ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
10198, 100mpbid 147 . . . . . . 7 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
102101ex 115 . . . . . 6 (((𝜑𝑟Q) ∧ 𝑎N) → ((𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
103102rexlimdva 2594 . . . . 5 ((𝜑𝑟Q) → (∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
104103expimpd 363 . . . 4 (𝜑 → ((𝑟Q ∧ ∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
10514, 104biimtrid 152 . . 3 (𝜑 → (𝑟 ∈ {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)} → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
106105ssrdv 3161 . 2 (𝜑 → {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
10710, 106eqsstrid 3201 1 (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
Colors of variables: wff set class
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 3129  cop 3595   class class class wbr 4003  wf 5212  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  1oc1o 6409  [cec 6532  Ncnpi 7270   <N clti 7273   ~Q ceq 7277  Qcnq 7278   +Q cplq 7280  *Qcrq 7282   <Q cltq 7283  Pcnp 7289   +P cpp 7291
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468
This theorem is referenced by:  caucvgprlem1  7677
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