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Theorem caucvgprprlem2 6962
Description: Lemma for caucvgprpr 6964. 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 6935 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
4 ltrelpi 6576 . . . . . . . . . 10 <N ⊆ (N × N)
54brel 4418 . . . . . . . . 9 (𝐽 <N 𝐾 → (𝐽N𝐾N))
61, 5syl 14 . . . . . . . 8 (𝜑 → (𝐽N𝐾N))
76simprd 112 . . . . . . 7 (𝜑𝐾N)
8 nnnq 6674 . . . . . . . 8 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
9 recclnq 6644 . . . . . . . 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 6799 . . . . . 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 5335 . . . . 5 (𝜑 → (𝐹𝐾) ∈ P)
17 ltaprg 6871 . . . . 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 1170 . . . 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 6789 . . . . 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 6789 . . . . 5 (((𝐹𝐾) ∈ P𝑄P) → ((𝐹𝐾) +P 𝑄) ∈ P)
2316, 14, 22syl2anc 403 . . . 4 (𝜑 → ((𝐹𝐾) +P 𝑄) ∈ P)
24 ltdfpr 6758 . . . 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 3796 . . . . . . . . . . . 12 (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
3130cbvabv 2203 . . . . . . . . . . 11 {𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}
32 breq2 3797 . . . . . . . . . . . 12 (𝑢 = 𝑞 → ((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞))
3332cbvabv 2203 . . . . . . . . . . 11 {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}
3431, 33opeq12i 3583 . . . . . . . . . 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 5554 . . . . . . . . 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 5212 . . . . . . . 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 2172 . . . . . . 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 6799 . . . . . . . . . . 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 6789 . . . . . . . . . 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 6804 . . . . . . . 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 5209 . . . . . . . . 9 (𝑟 = 𝐾 → (𝐹𝑟) = (𝐹𝐾))
47 opeq1 3578 . . . . . . . . . . . . . 14 (𝑟 = 𝐾 → ⟨𝑟, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
4847eceq1d 6208 . . . . . . . . . . . . 13 (𝑟 = 𝐾 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
4948fveq2d 5213 . . . . . . . . . . . 12 (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
5049breq2d 3805 . . . . . . . . . . 11 (𝑟 = 𝐾 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
5150abbidv 2197 . . . . . . . . . 10 (𝑟 = 𝐾 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )})
5249breq1d 3803 . . . . . . . . . . 11 (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞))
5352abbidv 2197 . . . . . . . . . 10 (𝑟 = 𝐾 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞})
5451, 53opeq12d 3586 . . . . . . . . 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 5561 . . . . . . . 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 3803 . . . . . . 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 2702 . . . . . 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 3797 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑝 <Q 𝑢𝑝 <Q 𝑥))
6059abbidv 2197 . . . . . . . . 9 (𝑢 = 𝑥 → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q 𝑥})
61 breq1 3796 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 <Q 𝑞𝑥 <Q 𝑞))
6261abbidv 2197 . . . . . . . . 9 (𝑢 = 𝑥 → {𝑞𝑢 <Q 𝑞} = {𝑞𝑥 <Q 𝑞})
6360, 62opeq12d 3586 . . . . . . . 8 (𝑢 = 𝑥 → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
6463breq2d 3805 . . . . . . 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 2370 . . . . . 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 5212 . . . . . . 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 6615 . . . . . . . . 9 Q ∈ V
6968rabex 3930 . . . . . . . 8 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
7068rabex 3930 . . . . . . . 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 5805 . . . . . . 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 2102 . . . . . 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 2752 . . . . 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 2413 . . . 4 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄)))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘((𝐹𝐾) +P 𝑄))))
7727, 74, 75, 76syl12anc 1168 . . 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 6956 . . . . 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 6758 . . . 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 2482 1 (𝜑𝐿<P ((𝐹𝐾) +P 𝑄))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  {cab 2068  wral 2349  wrex 2350  {crab 2353  cop 3409   class class class wbr 3793  wf 4928  cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  1𝑜c1o 6058  [cec 6170  Ncnpi 6524   <N clti 6527   ~Q ceq 6531  Qcnq 6532   +Q cplq 6534  *Qcrq 6536   <Q cltq 6537  Pcnp 6543   +P cpp 6545  <P cltp 6547
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprprlemlim  6963
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