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Theorem caucvgprprlemclphr 7477
 Description: Lemma for caucvgprpr 7484. The putative limit is a positive real. Like caucvgprprlemcl 7476 but without a distinct variable constraint between 𝜑 and 𝑟. (Contributed by Jim Kingdon, 19-Jun-2021.)
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}⟩}⟩
Assertion
Ref Expression
caucvgprprlemclphr (𝜑𝐿P)
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟   𝐹,𝑙,𝑢,𝑟,𝑘   𝑛,𝐹,𝑘   𝑘,𝐿   𝑢,𝑙,𝑝,𝑞,𝑟   𝑚,𝑟   𝑘,𝑝,𝑞,𝑟   𝑢,𝑛,𝑙,𝑘
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemclphr
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . 2 (𝜑𝐹:NP)
2 caucvgprpr.cau . 2 (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}⟩))))
3 caucvgprpr.bnd . 2 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 caucvgprpr.lim . . 3 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}⟩}⟩
5 opeq1 3673 . . . . . . . . . . . . . 14 (𝑟 = 𝑠 → ⟨𝑟, 1o⟩ = ⟨𝑠, 1o⟩)
65eceq1d 6431 . . . . . . . . . . . . 13 (𝑟 = 𝑠 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑠, 1o⟩] ~Q )
76fveq2d 5391 . . . . . . . . . . . 12 (𝑟 = 𝑠 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑠, 1o⟩] ~Q ))
87oveq2d 5756 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )))
98breq2d 3909 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))))
109abbidv 2233 . . . . . . . . 9 (𝑟 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))})
118breq1d 3907 . . . . . . . . . 10 (𝑟 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞))
1211abbidv 2233 . . . . . . . . 9 (𝑟 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞})
1310, 12opeq12d 3681 . . . . . . . 8 (𝑟 = 𝑠 → ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩)
14 fveq2 5387 . . . . . . . 8 (𝑟 = 𝑠 → (𝐹𝑟) = (𝐹𝑠))
1513, 14breq12d 3910 . . . . . . 7 (𝑟 = 𝑠 → (⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑠)))
1615cbvrexv 2630 . . . . . 6 (∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ∃𝑠N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑠))
1716a1i 9 . . . . 5 (𝑙Q → (∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ∃𝑠N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑠)))
1817rabbiia 2643 . . . 4 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} = {𝑙Q ∣ ∃𝑠N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑠)}
197breq2d 3909 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )))
2019abbidv 2233 . . . . . . . . . 10 (𝑟 = 𝑠 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )})
217breq1d 3907 . . . . . . . . . . 11 (𝑟 = 𝑠 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞))
2221abbidv 2233 . . . . . . . . . 10 (𝑟 = 𝑠 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞})
2320, 22opeq12d 3681 . . . . . . . . 9 (𝑟 = 𝑠 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)
2414, 23oveq12d 5758 . . . . . . . 8 (𝑟 = 𝑠 → ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩))
2524breq1d 3907 . . . . . . 7 (𝑟 = 𝑠 → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩))
2625cbvrexv 2630 . . . . . 6 (∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ∃𝑠N ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩)
2726a1i 9 . . . . 5 (𝑢Q → (∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ∃𝑠N ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩))
2827rabbiia 2643 . . . 4 {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 ∣ ∃𝑠N ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}
2918, 28opeq12i 3678 . . 3 ⟨{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}⟩}⟩ = ⟨{𝑙Q ∣ ∃𝑠N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑠)}, {𝑢Q ∣ ∃𝑠N ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
304, 29eqtri 2136 . 2 𝐿 = ⟨{𝑙Q ∣ ∃𝑠N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑠)}, {𝑢Q ∣ ∃𝑠N ((𝐹𝑠) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
311, 2, 3, 30caucvgprprlemcl 7476 1 (𝜑𝐿P)
 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  1oc1o 6272  [cec 6393  Ncnpi 7044
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