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Theorem caucvgprlemlim 7482
Description: Lemma for caucvgpr 7483. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemlim (𝜑 → ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑢,𝑙,𝑘   𝑛,𝐹,𝑘   𝑗,𝑘,𝜑,𝑥   𝑘,𝑙,𝑢,𝑥,𝑗   𝑗,𝐿,𝑘
Allowed substitution hints:   𝜑(𝑢,𝑛,𝑙)   𝐴(𝑥,𝑢,𝑘,𝑛,𝑙)   𝐹(𝑥)   𝐿(𝑥,𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlemlim
StepHypRef Expression
1 archrecnq 7464 . . . 4 (𝑥Q → ∃𝑗N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
21adantl 275 . . 3 ((𝜑𝑥Q) → ∃𝑗N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
3 caucvgpr.f . . . . . . . . . 10 (𝜑𝐹:NQ)
43ad5antr 487 . . . . . . . . 9 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝐹:NQ)
5 caucvgpr.cau . . . . . . . . . 10 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
65ad5antr 487 . . . . . . . . 9 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
7 caucvgpr.bnd . . . . . . . . . 10 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
87ad5antr 487 . . . . . . . . 9 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ∀𝑗N 𝐴 <Q (𝐹𝑗))
9 caucvgpr.lim . . . . . . . . 9 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
10 simpr 109 . . . . . . . . . 10 ((𝜑𝑥Q) → 𝑥Q)
1110ad4antr 485 . . . . . . . . 9 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝑥Q)
12 simpr 109 . . . . . . . . 9 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝑗 <N 𝑘)
13 simpllr 523 . . . . . . . . 9 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
144, 6, 8, 9, 11, 12, 13caucvgprlem1 7480 . . . . . . . 8 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → ⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩))
154, 6, 8, 9, 11, 12, 13caucvgprlem2 7481 . . . . . . . 8 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)
1614, 15jca 304 . . . . . . 7 ((((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) ∧ 𝑗 <N 𝑘) → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))
1716ex 114 . . . . . 6 (((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘N) → (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
1817ralrimiva 2503 . . . . 5 ((((𝜑𝑥Q) ∧ 𝑗N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) → ∀𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
1918ex 114 . . . 4 (((𝜑𝑥Q) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → ∀𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))))
2019reximdva 2532 . . 3 ((𝜑𝑥Q) → (∃𝑗N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))))
212, 20mpd 13 . 2 ((𝜑𝑥Q) → ∃𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
2221ralrimiva 2503 1 (𝜑 → ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  {cab 2123  wral 2414  wrex 2415  {crab 2418  cop 3525   class class class wbr 3924  wf 5114  cfv 5118  (class class class)co 5767  1oc1o 6299  [cec 6420  Ncnpi 7073   <N clti 7076   ~Q ceq 7080  Qcnq 7081   +Q cplq 7083  *Qcrq 7085   <Q cltq 7086   +P cpp 7094  <P cltp 7096
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269  df-iltp 7271
This theorem is referenced by:  caucvgpr  7483
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