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Theorem caucvgprprlemclphr 6801
Description: Lemma for caucvgprpr 6808. The putative limit is a positive real. Like caucvgprprlemcl 6800 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 3549 . . . . . . . . . . . . . 14 (𝑟 = 𝑠 → ⟨𝑟, 1𝑜⟩ = ⟨𝑠, 1𝑜⟩)
65eceq1d 6142 . . . . . . . . . . . . 13 (𝑟 = 𝑠 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑠, 1𝑜⟩] ~Q )
76fveq2d 5182 . . . . . . . . . . . 12 (𝑟 = 𝑠 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ))
87oveq2d 5528 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )))
98breq2d 3776 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ))))
109abbidv 2155 . . . . . . . . 9 (𝑟 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ))})
118breq1d 3774 . . . . . . . . . 10 (𝑟 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )) <Q 𝑞))
1211abbidv 2155 . . . . . . . . 9 (𝑟 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )) <Q 𝑞})
1310, 12opeq12d 3557 . . . . . . . 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 5178 . . . . . . . 8 (𝑟 = 𝑠 → (𝐹𝑟) = (𝐹𝑠))
1513, 14breq12d 3777 . . . . . . 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 2534 . . . . . 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 2547 . . . 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 3776 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )))
2019abbidv 2155 . . . . . . . . . 10 (𝑟 = 𝑠 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )})
217breq1d 3774 . . . . . . . . . . 11 (𝑟 = 𝑠 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ) <Q 𝑞))
2221abbidv 2155 . . . . . . . . . 10 (𝑟 = 𝑠 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ) <Q 𝑞})
2320, 22opeq12d 3557 . . . . . . . . 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 5530 . . . . . . . 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 3774 . . . . . . 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 2534 . . . . . 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 2547 . . . 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 3554 . . 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 2060 . 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 6800 1 (𝜑𝐿P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  {crab 2310  cop 3378   class class class wbr 3764  wf 4898  cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  Qcnq 6376   +Q cplq 6378  *Qcrq 6380   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  caucvgprprlemexbt  6802  caucvgprprlemexb  6803
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