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Theorem caucvgprlem2 7670
Description: Lemma for caucvgpr 7672. 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 7655 . . . 4 (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
4 caucvgpr.f . . . . 5 (𝜑𝐹:NQ)
5 ltrelpi 7314 . . . . . . . 8 <N ⊆ (N × N)
65brel 4675 . . . . . . 7 (𝐽 <N 𝐾 → (𝐽N𝐾N))
71, 6syl 14 . . . . . 6 (𝜑 → (𝐽N𝐾N))
87simprd 114 . . . . 5 (𝜑𝐾N)
94, 8ffvelcdmd 5648 . . . 4 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 7392 . . . 4 (((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
113, 9, 10syl2anc 411 . . 3 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 ltbtwnnqq 7405 . . 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 122 . 2 (𝜑 → ∃𝑥Q (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
14 simprl 529 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥Q)
158adantr 276 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐾N)
16 simprrl 539 . . . . . 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 5511 . . . . . . . . 9 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
18 opeq1 3776 . . . . . . . . . . 11 (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
1918eceq1d 6565 . . . . . . . . . 10 (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
2019fveq2d 5515 . . . . . . . . 9 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
2117, 20oveq12d 5887 . . . . . . . 8 (𝑗 = 𝐾 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
2221breq1d 4010 . . . . . . 7 (𝑗 = 𝐾 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥 ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥))
2322rspcev 2841 . . . . . 6 ((𝐾N ∧ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥)
2415, 16, 23syl2anc 411 . . . . 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 4004 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
2625rexbidv 2478 . . . . . 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 5514 . . . . . . 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 7353 . . . . . . . . 9 Q ∈ V
3029rabex 4144 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
3129rabex 4144 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
3230, 31op2nd 6142 . . . . . . 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 2198 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
3426, 33elrab2 2896 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
3514, 24, 34sylanbrc 417 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ (2nd𝐿))
36 simprrr 540 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
37 vex 2740 . . . . . . 7 𝑥 ∈ V
38 breq1 4003 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
3937, 38elab 2881 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
4036, 39sylibr 134 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)})
41 ltnqex 7539 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ∈ V
42 gtnqex 7540 . . . . . 6 {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢} ∈ V
4341, 42op1st 6141 . . . . 5 (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩) = {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}
4440, 43eleqtrrdi 2271 . . . 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 2526 . . . 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 1236 . . 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 7666 . . . . 5 (𝜑𝐿P)
5049adantr 276 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐿P)
51 caucvgprlemlim.q . . . . . . 7 (𝜑𝑄Q)
52 addclnq 7365 . . . . . . 7 (((𝐹𝐾) ∈ Q𝑄Q) → ((𝐹𝐾) +Q 𝑄) ∈ Q)
539, 51, 52syl2anc 411 . . . . . 6 (𝜑 → ((𝐹𝐾) +Q 𝑄) ∈ Q)
54 nqprlu 7537 . . . . . 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 276 . . . 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 7496 . . . 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 411 . . 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 167 . 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 2599 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 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  cop 3594   class class class wbr 4000  wf 5208  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  1oc1o 6404  [cec 6527  Ncnpi 7262   <N clti 7265   ~Q ceq 7269  Qcnq 7270   +Q cplq 7272  *Qcrq 7274   <Q cltq 7275  Pcnp 7281  <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  caucvgprlemlim  7671
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