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Theorem caucvgprprlemlim 7542
Description: Lemma for caucvgprpr 7543. 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 7495 . . . 4 (𝑥P → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
21adantl 275 . . 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 488 . . . . . . . . 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 488 . . . . . . . . 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 488 . . . . . . . . 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 109 . . . . . . . . . 10 ((𝜑𝑥P) → 𝑥P)
1110ad4antr 486 . . . . . . . . 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 109 . . . . . . . . 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 524 . . . . . . . . 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 7540 . . . . . . . 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 7541 . . . . . . . 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 304 . . . . . . 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 114 . . . . . 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 2508 . . . . 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 114 . . . 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 2537 . . 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 2508 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 103   = wceq 1332  wcel 1481  {cab 2126  wral 2417  wrex 2418  {crab 2421  cop 3534   class class class wbr 3936  wf 5126  cfv 5130  (class class class)co 5781  1oc1o 6313  [cec 6434  Ncnpi 7103   <N clti 7106   ~Q ceq 7110  Qcnq 7111   +Q cplq 7113  *Qcrq 7115   <Q cltq 7116  Pcnp 7122   +P cpp 7124  <P cltp 7126
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-iplp 7299  df-iltp 7301
This theorem is referenced by:  caucvgprpr  7543
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