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Theorem caucvgprprlemlim 7214
Description: Lemma for caucvgprpr 7215. The putative limit is a limit. (Contributed by Jim Kingdon, 21-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}⟩}⟩
Assertion
Ref Expression
caucvgprprlemlim (𝜑 → ∀𝑥P𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑗   𝑢,𝐹,𝑟,𝑙,𝑘,𝑛   𝜑,𝑘,𝑟   𝑘,𝐿   𝑗,𝑘,𝜑,𝑥   𝑘,𝑙,𝑢,𝑝,𝑞,𝑟   𝑗,𝑟,𝑥   𝑞,𝑙,𝑟   𝑢,𝑝,𝑞,𝑟   𝑚,𝑟   𝑘,𝑛,𝑢,𝑙   𝑗,𝑙,𝑢   𝑛,𝑟
Allowed substitution hints:   𝜑(𝑢,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑥,𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑥,𝑗,𝑞,𝑝)   𝐿(𝑥,𝑢,𝑗,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemlim
StepHypRef Expression
1 archrecpr 7167 . . . 4 (𝑥P → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
21adantl 271 . . 3 ((𝜑𝑥P) → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
3 caucvgprpr.f . . . . . . . . . 10 (𝜑𝐹:NP)
43ad5antr 480 . . . . . . . . 9 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝐹:NP)
5 caucvgprpr.cau . . . . . . . . . 10 (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}⟩))))
65ad5antr 480 . . . . . . . . 9 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ∀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}⟩))))
7 caucvgprpr.bnd . . . . . . . . . 10 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
87ad5antr 480 . . . . . . . . 9 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ∀𝑚N 𝐴<P (𝐹𝑚))
9 caucvgprpr.lim . . . . . . . . 9 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}⟩}⟩
10 simpr 108 . . . . . . . . . 10 ((𝜑𝑥P) → 𝑥P)
1110ad4antr 478 . . . . . . . . 9 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝑥P)
12 simpr 108 . . . . . . . . 9 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝑗 <N 𝑘)
13 simpllr 501 . . . . . . . . 9 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
144, 6, 8, 9, 11, 12, 13caucvgprprlem1 7212 . . . . . . . 8 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → (𝐹𝑘)<P (𝐿 +P 𝑥))
154, 6, 8, 9, 11, 12, 13caucvgprprlem2 7213 . . . . . . . 8 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝐿<P ((𝐹𝑘) +P 𝑥))
1614, 15jca 300 . . . . . . 7 ((((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥)))
1716ex 113 . . . . . 6 (((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘N) → (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))
1817ralrimiva 2442 . . . . 5 ((((𝜑𝑥P) ∧ 𝑗N) ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) → ∀𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))
1918ex 113 . . . 4 (((𝜑𝑥P) ∧ 𝑗N) → (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥 → ∀𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥)))))
2019reximdva 2471 . . 3 ((𝜑𝑥P) → (∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥)))))
212, 20mpd 13 . 2 ((𝜑𝑥P) → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))
2221ralrimiva 2442 1 (𝜑 → ∀𝑥P𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  {cab 2071  wral 2355  wrex 2356  {crab 2359  cop 3434   class class class wbr 3820  wf 4977  cfv 4981  (class class class)co 5613  1𝑜c1o 6128  [cec 6242  Ncnpi 6775   <N clti 6778   ~Q ceq 6782  Qcnq 6783   +Q cplq 6785  *Qcrq 6787   <Q cltq 6788  Pcnp 6794   +P cpp 6796  <P cltp 6798
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-iplp 6971  df-iltp 6973
This theorem is referenced by:  caucvgprpr  7215
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