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Theorem caucvgprlem2 7430
Description: Lemma for caucvgpr 7432. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-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}⟩
caucvgprlemlim.q (𝜑𝑄Q)
caucvgprlemlim.jk (𝜑𝐽 <N 𝐾)
caucvgprlemlim.jkq (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)
Assertion
Ref Expression
caucvgprlem2 (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑢,𝑙   𝑛,𝐹,𝑘   𝑗,𝐾,𝑢,𝑙   𝑗,𝐿,𝑘   𝑄,𝑙,𝑢   𝑗,𝑙   𝑗,𝑘   𝑘,𝑛
Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝑄(𝑗,𝑘,𝑛)   𝐽(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐾(𝑘,𝑛)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 caucvgprlemlim.jk . . . . 5 (𝜑𝐽 <N 𝐾)
2 caucvgprlemlim.jkq . . . . 5 (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)
31, 2caucvgprlemk 7415 . . . 4 (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
4 caucvgpr.f . . . . 5 (𝜑𝐹:NQ)
5 ltrelpi 7074 . . . . . . . 8 <N ⊆ (N × N)
65brel 4549 . . . . . . 7 (𝐽 <N 𝐾 → (𝐽N𝐾N))
71, 6syl 14 . . . . . 6 (𝜑 → (𝐽N𝐾N))
87simprd 113 . . . . 5 (𝜑𝐾N)
94, 8ffvelrnd 5508 . . . 4 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 7152 . . . 4 (((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
113, 9, 10syl2anc 406 . . 3 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 ltbtwnnqq 7165 . . 3 (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄) ↔ ∃𝑥Q (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
1311, 12sylib 121 . 2 (𝜑 → ∃𝑥Q (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
14 simprl 503 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥Q)
158adantr 272 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐾N)
16 simprrl 511 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥)
17 fveq2 5373 . . . . . . . . 9 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
18 opeq1 3669 . . . . . . . . . . 11 (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
1918eceq1d 6417 . . . . . . . . . 10 (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
2019fveq2d 5377 . . . . . . . . 9 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
2117, 20oveq12d 5744 . . . . . . . 8 (𝑗 = 𝐾 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
2221breq1d 3903 . . . . . . 7 (𝑗 = 𝐾 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥 ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥))
2322rspcev 2758 . . . . . 6 ((𝐾N ∧ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥)
2415, 16, 23syl2anc 406 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥)
25 breq2 3897 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
2625rexbidv 2410 . . . . . 6 (𝑢 = 𝑥 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
27 caucvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
2827fveq2i 5376 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
29 nqex 7113 . . . . . . . . 9 Q ∈ V
3029rabex 4030 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
3129rabex 4030 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
3230, 31op2nd 5997 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
3328, 32eqtri 2133 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
3426, 33elrab2 2810 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
3514, 24, 34sylanbrc 411 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ (2nd𝐿))
36 simprrr 512 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
37 vex 2658 . . . . . . 7 𝑥 ∈ V
38 breq1 3896 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
3937, 38elab 2796 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
4036, 39sylibr 133 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)})
41 ltnqex 7299 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ∈ V
42 gtnqex 7300 . . . . . 6 {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢} ∈ V
4341, 42op1st 5996 . . . . 5 (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩) = {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}
4440, 43syl6eleqr 2206 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩))
45 rspe 2453 . . . 4 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)))
4614, 35, 44, 45syl12anc 1195 . . 3 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)))
47 caucvgpr.cau . . . . . 6 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
48 caucvgpr.bnd . . . . . 6 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
494, 47, 48, 27caucvgprlemcl 7426 . . . . 5 (𝜑𝐿P)
5049adantr 272 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐿P)
51 caucvgprlemlim.q . . . . . . 7 (𝜑𝑄Q)
52 addclnq 7125 . . . . . . 7 (((𝐹𝐾) ∈ Q𝑄Q) → ((𝐹𝐾) +Q 𝑄) ∈ Q)
539, 51, 52syl2anc 406 . . . . . 6 (𝜑 → ((𝐹𝐾) +Q 𝑄) ∈ Q)
54 nqprlu 7297 . . . . . 6 (((𝐹𝐾) +Q 𝑄) ∈ Q → ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P)
5553, 54syl 14 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P)
5655adantr 272 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P)
57 ltdfpr 7256 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P) → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩))))
5850, 56, 57syl2anc 406 . . 3 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩))))
5946, 58mpbird 166 . 2 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)
6013, 59rexlimddv 2526 1 (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1312  wcel 1461  {cab 2099  wral 2388  wrex 2389  {crab 2392  cop 3494   class class class wbr 3893  wf 5075  cfv 5079  (class class class)co 5726  1st c1st 5988  2nd c2nd 5989  1oc1o 6258  [cec 6379  Ncnpi 7022   <N clti 7025   ~Q ceq 7029  Qcnq 7030   +Q cplq 7032  *Qcrq 7034   <Q cltq 7035  Pcnp 7041  <P cltp 7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-inp 7216  df-iltp 7220
This theorem is referenced by:  caucvgprlemlim  7431
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