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Theorem caucvgprprlemclphr 6993
Description: Lemma for caucvgprpr 7000. The putative limit is a positive real. Like caucvgprprlemcl 6992 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 3591 . . . . . . . . . . . . . 14 (𝑟 = 𝑠 → ⟨𝑟, 1𝑜⟩ = ⟨𝑠, 1𝑜⟩)
65eceq1d 6230 . . . . . . . . . . . . 13 (𝑟 = 𝑠 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑠, 1𝑜⟩] ~Q )
76fveq2d 5234 . . . . . . . . . . . 12 (𝑟 = 𝑠 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ))
87oveq2d 5580 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )))
98breq2d 3818 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ))))
109abbidv 2200 . . . . . . . . 9 (𝑟 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ))})
118breq1d 3816 . . . . . . . . . 10 (𝑟 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )) <Q 𝑞))
1211abbidv 2200 . . . . . . . . 9 (𝑟 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )) <Q 𝑞})
1310, 12opeq12d 3599 . . . . . . . 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 5230 . . . . . . . 8 (𝑟 = 𝑠 → (𝐹𝑟) = (𝐹𝑠))
1513, 14breq12d 3819 . . . . . . 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 2583 . . . . . 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 2596 . . . 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 3818 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )))
2019abbidv 2200 . . . . . . . . . 10 (𝑟 = 𝑠 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑠, 1𝑜⟩] ~Q )})
217breq1d 3816 . . . . . . . . . . 11 (𝑟 = 𝑠 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ) <Q 𝑞))
2221abbidv 2200 . . . . . . . . . 10 (𝑟 = 𝑠 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑠, 1𝑜⟩] ~Q ) <Q 𝑞})
2320, 22opeq12d 3599 . . . . . . . . 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 5582 . . . . . . . 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 3816 . . . . . . 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 2583 . . . . . 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 2596 . . . 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 3596 . . 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 2103 . 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 6992 1 (𝜑𝐿P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  {crab 2357  cop 3420   class class class wbr 3806  wf 4949  cfv 4953  (class class class)co 5564  1𝑜c1o 6079  [cec 6192  Ncnpi 6560   <N clti 6563   ~Q ceq 6567  Qcnq 6568   +Q cplq 6570  *Qcrq 6572   <Q cltq 6573  Pcnp 6579   +P cpp 6581  <P cltp 6583
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 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-enq0 6712  df-nq0 6713  df-0nq0 6714  df-plq0 6715  df-mq0 6716  df-inp 6754  df-iplp 6756  df-iltp 6758
This theorem is referenced by:  caucvgprprlemexbt  6994  caucvgprprlemexb  6995
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