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Theorem caucvgprprlem2 7482
 Description: Lemma for caucvgprpr 7484. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
caucvgprprlemlim.q (𝜑𝑄P)
caucvgprprlemlim.jk (𝜑𝐽 <N 𝐾)
caucvgprprlemlim.jkq (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
Assertion
Ref Expression
caucvgprprlem2 (𝜑𝐿<P ((𝐹𝐾) +P 𝑄))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟   𝐹,𝑟,𝑢,𝑙,𝑘   𝑛,𝐹   𝐾,𝑙,𝑝,𝑢,𝑞,𝑟   𝐽,𝑙,𝑢   𝑘,𝐿   𝜑,𝑟   𝑘,𝑛   𝑘,𝑟   𝑞,𝑙,𝑟   𝑚,𝑟   𝑘,𝑝,𝑞   𝑢,𝑛,𝑙,𝑘
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑄(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑞,𝑝)   𝐽(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝)   𝐾(𝑘,𝑚,𝑛)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemlim.jk . . . . 5 (𝜑𝐽 <N 𝐾)
2 caucvgprprlemlim.jkq . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
31, 2caucvgprprlemk 7455 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
4 ltrelpi 7096 . . . . . . . . . 10 <N ⊆ (N × N)
54brel 4559 . . . . . . . . 9 (𝐽 <N 𝐾 → (𝐽N𝐾N))
61, 5syl 14 . . . . . . . 8 (𝜑 → (𝐽N𝐾N))
76simprd 113 . . . . . . 7 (𝜑𝐾N)
8 nnnq 7194 . . . . . . . 8 (𝐾N → [⟨𝐾, 1o⟩] ~QQ)
9 recclnq 7164 . . . . . . . 8 ([⟨𝐾, 1o⟩] ~QQ → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
108, 9syl 14 . . . . . . 7 (𝐾N → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
117, 10syl 14 . . . . . 6 (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
12 nqprlu 7319 . . . . . 6 ((*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
1311, 12syl 14 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
14 caucvgprprlemlim.q . . . . 5 (𝜑𝑄P)
15 caucvgprpr.f . . . . . 6 (𝜑𝐹:NP)
1615, 7ffvelrnd 5522 . . . . 5 (𝜑 → (𝐹𝐾) ∈ P)
17 ltaprg 7391 . . . . 5 ((⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P𝑄P ∧ (𝐹𝐾) ∈ P) → (⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄)))
1813, 14, 16, 17syl3anc 1199 . . . 4 (𝜑 → (⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄)))
193, 18mpbid 146 . . 3 (𝜑 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄))
20 addclpr 7309 . . . . 5 (((𝐹𝐾) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
2116, 13, 20syl2anc 406 . . . 4 (𝜑 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
22 addclpr 7309 . . . . 5 (((𝐹𝐾) ∈ P𝑄P) → ((𝐹𝐾) +P 𝑄) ∈ P)
2316, 14, 22syl2anc 406 . . . 4 (𝜑 → ((𝐹𝐾) +P 𝑄) ∈ P)
24 ltdfpr 7278 . . . 4 ((((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P ∧ ((𝐹𝐾) +P 𝑄) ∈ P) → (((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄) ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))))
2521, 23, 24syl2anc 406 . . 3 (𝜑 → (((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄) ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))))
2619, 25mpbid 146 . 2 (𝜑 → ∃𝑥Q (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))
27 simprl 503 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝑥Q)
287adantr 272 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝐾N)
29 simprrl 511 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)))
30 breq1 3900 . . . . . . . . . . . 12 (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
3130cbvabv 2239 . . . . . . . . . . 11 {𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}
32 breq2 3901 . . . . . . . . . . . 12 (𝑢 = 𝑞 → ((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞))
3332cbvabv 2239 . . . . . . . . . . 11 {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}
3431, 33opeq12i 3678 . . . . . . . . . 10 ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩
3534oveq2i 5751 . . . . . . . . 9 ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)
3635fveq2i 5390 . . . . . . . 8 (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) = (2nd ‘((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
3729, 36syl6eleq 2208 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)))
38 nqprlu 7319 . . . . . . . . . . 11 ((*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
3911, 38syl 14 . . . . . . . . . 10 (𝜑 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
40 addclpr 7309 . . . . . . . . . 10 (((𝐹𝐾) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4116, 39, 40syl2anc 406 . . . . . . . . 9 (𝜑 → ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4241adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
43 nqpru 7324 . . . . . . . 8 ((𝑥Q ∧ ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
4427, 42, 43syl2anc 406 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
4537, 44mpbid 146 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
46 fveq2 5387 . . . . . . . . 9 (𝑟 = 𝐾 → (𝐹𝑟) = (𝐹𝐾))
47 opeq1 3673 . . . . . . . . . . . . . 14 (𝑟 = 𝐾 → ⟨𝑟, 1o⟩ = ⟨𝐾, 1o⟩)
4847eceq1d 6431 . . . . . . . . . . . . 13 (𝑟 = 𝐾 → [⟨𝑟, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
4948fveq2d 5391 . . . . . . . . . . . 12 (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
5049breq2d 3909 . . . . . . . . . . 11 (𝑟 = 𝐾 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
5150abbidv 2233 . . . . . . . . . 10 (𝑟 = 𝐾 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )})
5249breq1d 3907 . . . . . . . . . . 11 (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞))
5352abbidv 2233 . . . . . . . . . 10 (𝑟 = 𝐾 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞})
5451, 53opeq12d 3681 . . . . . . . . 9 (𝑟 = 𝐾 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)
5546, 54oveq12d 5758 . . . . . . . 8 (𝑟 = 𝐾 → ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
5655breq1d 3907 . . . . . . 7 (𝑟 = 𝐾 → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ ↔ ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
5756rspcev 2761 . . . . . 6 ((𝐾N ∧ ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩) → ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
5828, 45, 57syl2anc 406 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
59 breq2 3901 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑝 <Q 𝑢𝑝 <Q 𝑥))
6059abbidv 2233 . . . . . . . . 9 (𝑢 = 𝑥 → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q 𝑥})
61 breq1 3900 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 <Q 𝑞𝑥 <Q 𝑞))
6261abbidv 2233 . . . . . . . . 9 (𝑢 = 𝑥 → {𝑞𝑢 <Q 𝑞} = {𝑞𝑥 <Q 𝑞})
6360, 62opeq12d 3681 . . . . . . . 8 (𝑢 = 𝑥 → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
6463breq2d 3909 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
6564rexbidv 2413 . . . . . 6 (𝑢 = 𝑥 → (∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
66 caucvgprpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
6766fveq2i 5390 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩)
68 nqex 7135 . . . . . . . . 9 Q ∈ V
6968rabex 4040 . . . . . . . 8 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
7068rabex 4040 . . . . . . . 8 {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ∈ V
7169, 70op2nd 6011 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩) = {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}
7267, 71eqtri 2136 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}
7365, 72elrab2 2814 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
7427, 58, 73sylanbrc 411 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd𝐿))
75 simprrr 512 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))
76 rspe 2456 . . . 4 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))
7727, 74, 75, 76syl12anc 1197 . . 3 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))
78 caucvgprpr.cau . . . . . 6 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
79 caucvgprpr.bnd . . . . . 6 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
8015, 78, 79, 66caucvgprprlemcl 7476 . . . . 5 (𝜑𝐿P)
8180adantr 272 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝐿P)
8223adantr 272 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → ((𝐹𝐾) +P 𝑄) ∈ P)
83 ltdfpr 7278 . . . 4 ((𝐿P ∧ ((𝐹𝐾) +P 𝑄) ∈ P) → (𝐿<P ((𝐹𝐾) +P 𝑄) ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))))
8481, 82, 83syl2anc 406 . . 3 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → (𝐿<P ((𝐹𝐾) +P 𝑄) ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))))
8577, 84mpbird 166 . 2 ((𝜑 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩)) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))) → 𝐿<P ((𝐹𝐾) +P 𝑄))
8626, 85rexlimddv 2529 1 (𝜑𝐿<P ((𝐹𝐾) +P 𝑄))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2391  ∃wrex 2392  {crab 2395  ⟨cop 3498   class class class wbr 3897  ⟶wf 5087  ‘cfv 5091  (class class class)co 5740  1st c1st 6002  2nd c2nd 6003  1oc1o 6272  [cec 6393  Ncnpi 7044
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