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Theorem caucvgprlem2 7002
Description: Lemma for caucvgpr 7004. 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 6987 . . . 4 (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄)
4 caucvgpr.f . . . . 5 (𝜑𝐹:NQ)
5 ltrelpi 6646 . . . . . . . 8 <N ⊆ (N × N)
65brel 4438 . . . . . . 7 (𝐽 <N 𝐾 → (𝐽N𝐾N))
71, 6syl 14 . . . . . 6 (𝜑 → (𝐽N𝐾N))
87simprd 112 . . . . 5 (𝜑𝐾N)
94, 8ffvelrnd 5356 . . . 4 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 6724 . . . 4 (((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
113, 9, 10syl2anc 403 . . 3 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 ltbtwnnqq 6737 . . 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 120 . 2 (𝜑 → ∃𝑥Q (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
14 simprl 498 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥Q)
158adantr 270 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐾N)
16 simprrl 506 . . . . . 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 5230 . . . . . . . . 9 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
18 opeq1 3590 . . . . . . . . . . 11 (𝑗 = 𝐾 → ⟨𝑗, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
1918eceq1d 6230 . . . . . . . . . 10 (𝑗 = 𝐾 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
2019fveq2d 5234 . . . . . . . . 9 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
2117, 20oveq12d 5582 . . . . . . . 8 (𝑗 = 𝐾 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
2221breq1d 3815 . . . . . . 7 (𝑗 = 𝐾 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑥 ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥))
2322rspcev 2710 . . . . . 6 ((𝐾N ∧ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑥)
2415, 16, 23syl2anc 403 . . . . 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 3809 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑥))
2625rexbidv 2374 . . . . . 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 5233 . . . . . . 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 6685 . . . . . . . . 9 Q ∈ V
3029rabex 3942 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
3129rabex 3942 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
3230, 31op2nd 5826 . . . . . . 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 2103 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
3426, 33elrab2 2760 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑥))
3514, 24, 34sylanbrc 408 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ (2nd𝐿))
36 simprrr 507 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
37 vex 2613 . . . . . . 7 𝑥 ∈ V
38 breq1 3808 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
3937, 38elab 2746 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
4036, 39sylibr 132 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)})
41 ltnqex 6871 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ∈ V
42 gtnqex 6872 . . . . . 6 {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢} ∈ V
4341, 42op1st 5825 . . . . 5 (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩) = {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}
4440, 43syl6eleqr 2176 . . . 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 2417 . . . 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 1168 . . 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 6998 . . . . 5 (𝜑𝐿P)
5049adantr 270 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐿P)
51 caucvgprlemlim.q . . . . . . 7 (𝜑𝑄Q)
52 addclnq 6697 . . . . . . 7 (((𝐹𝐾) ∈ Q𝑄Q) → ((𝐹𝐾) +Q 𝑄) ∈ Q)
539, 51, 52syl2anc 403 . . . . . 6 (𝜑 → ((𝐹𝐾) +Q 𝑄) ∈ Q)
54 nqprlu 6869 . . . . . 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 270 . . . 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 6828 . . . 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 403 . . 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 165 . 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 2486 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 102  wb 103   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  {crab 2357  cop 3419   class class class wbr 3805  wf 4948  cfv 4952  (class class class)co 5564  1st c1st 5817  2nd c2nd 5818  1𝑜c1o 6079  [cec 6192  Ncnpi 6594   <N clti 6597   ~Q ceq 6601  Qcnq 6602   +Q cplq 6604  *Qcrq 6606   <Q cltq 6607  Pcnp 6613  <P cltp 6617
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-pli 6627  df-mi 6628  df-lti 6629  df-plpq 6666  df-mpq 6667  df-enq 6669  df-nqqs 6670  df-plqqs 6671  df-mqqs 6672  df-1nqqs 6673  df-rq 6674  df-ltnqqs 6675  df-inp 6788  df-iltp 6792
This theorem is referenced by:  caucvgprlemlim  7003
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