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Theorem caucvgprprlem2 7248
Description: Lemma for caucvgprpr 7250. 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 7221 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
4 ltrelpi 6862 . . . . . . . . . 10 <N ⊆ (N × N)
54brel 4478 . . . . . . . . 9 (𝐽 <N 𝐾 → (𝐽N𝐾N))
61, 5syl 14 . . . . . . . 8 (𝜑 → (𝐽N𝐾N))
76simprd 112 . . . . . . 7 (𝜑𝐾N)
8 nnnq 6960 . . . . . . . 8 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
9 recclnq 6930 . . . . . . . 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 7085 . . . . . 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 5419 . . . . 5 (𝜑 → (𝐹𝐾) ∈ P)
17 ltaprg 7157 . . . . 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 1174 . . . 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 145 . . 3 (𝜑 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄))
20 addclpr 7075 . . . . 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 403 . . . 4 (𝜑 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
22 addclpr 7075 . . . . 5 (((𝐹𝐾) ∈ P𝑄P) → ((𝐹𝐾) +P 𝑄) ∈ P)
2316, 14, 22syl2anc 403 . . . 4 (𝜑 → ((𝐹𝐾) +P 𝑄) ∈ P)
24 ltdfpr 7044 . . . 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 403 . . 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 145 . 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 498 . . . 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 270 . . . . . 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 506 . . . . . . . 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 3840 . . . . . . . . . . . 12 (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
3130cbvabv 2211 . . . . . . . . . . 11 {𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}
32 breq2 3841 . . . . . . . . . . . 12 (𝑢 = 𝑞 → ((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞))
3332cbvabv 2211 . . . . . . . . . . 11 {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}
3431, 33opeq12i 3622 . . . . . . . . . 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 5645 . . . . . . . . 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 5292 . . . . . . . 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 2180 . . . . . . 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 7085 . . . . . . . . . . 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 7075 . . . . . . . . . 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 403 . . . . . . . . 9 (𝜑 → ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4241adantr 270 . . . . . . . 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 7090 . . . . . . . 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 403 . . . . . . 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 145 . . . . . 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 5289 . . . . . . . . 9 (𝑟 = 𝐾 → (𝐹𝑟) = (𝐹𝐾))
47 opeq1 3617 . . . . . . . . . . . . . 14 (𝑟 = 𝐾 → ⟨𝑟, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
4847eceq1d 6308 . . . . . . . . . . . . 13 (𝑟 = 𝐾 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
4948fveq2d 5293 . . . . . . . . . . . 12 (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
5049breq2d 3849 . . . . . . . . . . 11 (𝑟 = 𝐾 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
5150abbidv 2205 . . . . . . . . . 10 (𝑟 = 𝐾 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )})
5249breq1d 3847 . . . . . . . . . . 11 (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞))
5352abbidv 2205 . . . . . . . . . 10 (𝑟 = 𝐾 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞})
5451, 53opeq12d 3625 . . . . . . . . 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 5652 . . . . . . . 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 3847 . . . . . . 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 2722 . . . . . 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 403 . . . . 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 3841 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑝 <Q 𝑢𝑝 <Q 𝑥))
6059abbidv 2205 . . . . . . . . 9 (𝑢 = 𝑥 → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q 𝑥})
61 breq1 3840 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 <Q 𝑞𝑥 <Q 𝑞))
6261abbidv 2205 . . . . . . . . 9 (𝑢 = 𝑥 → {𝑞𝑢 <Q 𝑞} = {𝑞𝑥 <Q 𝑞})
6360, 62opeq12d 3625 . . . . . . . 8 (𝑢 = 𝑥 → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
6463breq2d 3849 . . . . . . 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 2381 . . . . . 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 5292 . . . . . . 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 6901 . . . . . . . . 9 Q ∈ V
6968rabex 3975 . . . . . . . 8 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
7068rabex 3975 . . . . . . . 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 5900 . . . . . . 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 2108 . . . . . 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 2772 . . . . 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 408 . . . 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 507 . . . 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 2424 . . . 4 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))
7727, 74, 75, 76syl12anc 1172 . . 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 7242 . . . . 5 (𝜑𝐿P)
8180adantr 270 . . . 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 270 . . . 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 7044 . . . 4 ((𝐿P ∧ ((𝐹𝐾) +P 𝑄) ∈ P) → (𝐿<P ((𝐹𝐾) +P 𝑄) ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))))
8481, 82, 83syl2anc 403 . . 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 165 . 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 2493 1 (𝜑𝐿<P ((𝐹𝐾) +P 𝑄))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  {crab 2363  cop 3444   class class class wbr 3837  wf 4998  cfv 5002  (class class class)co 5634  1st c1st 5891  2nd c2nd 5892  1𝑜c1o 6156  [cec 6270  Ncnpi 6810   <N clti 6813   ~Q ceq 6817  Qcnq 6818   +Q cplq 6820  *Qcrq 6822   <Q cltq 6823  Pcnp 6829   +P cpp 6831  <P cltp 6833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprprlemlim  7249
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