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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   <N clti 7047   ~Q ceq 7051  Qcnq 7052   +Q cplq 7054  *Qcrq 7056   <Q cltq 7057  Pcnp 7063   +P cpp 7065  <P cltp 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  caucvgprprlemexbt  7478  caucvgprprlemexb  7479
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