ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlem2 GIF version

Theorem caucvgprlem2 7642
Description: Lemma for caucvgpr 7644. 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 7627 . . . 4 (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
4 caucvgpr.f . . . . 5 (𝜑𝐹:NQ)
5 ltrelpi 7286 . . . . . . . 8 <N ⊆ (N × N)
65brel 4663 . . . . . . 7 (𝐽 <N 𝐾 → (𝐽N𝐾N))
71, 6syl 14 . . . . . 6 (𝜑 → (𝐽N𝐾N))
87simprd 113 . . . . 5 (𝜑𝐾N)
94, 8ffvelrnd 5632 . . . 4 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 7364 . . . 4 (((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
113, 9, 10syl2anc 409 . . 3 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 ltbtwnnqq 7377 . . 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 526 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥Q)
158adantr 274 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐾N)
16 simprrl 534 . . . . . 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 5496 . . . . . . . . 9 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
18 opeq1 3765 . . . . . . . . . . 11 (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
1918eceq1d 6549 . . . . . . . . . 10 (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
2019fveq2d 5500 . . . . . . . . 9 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
2117, 20oveq12d 5871 . . . . . . . 8 (𝑗 = 𝐾 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
2221breq1d 3999 . . . . . . 7 (𝑗 = 𝐾 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥 ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥))
2322rspcev 2834 . . . . . 6 ((𝐾N ∧ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥)
2415, 16, 23syl2anc 409 . . . . 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 3993 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
2625rexbidv 2471 . . . . . 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 5499 . . . . . . 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 7325 . . . . . . . . 9 Q ∈ V
3029rabex 4133 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
3129rabex 4133 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
3230, 31op2nd 6126 . . . . . . 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 2191 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
3426, 33elrab2 2889 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
3514, 24, 34sylanbrc 415 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 ∈ (2nd𝐿))
36 simprrr 535 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝑥 <Q ((𝐹𝐾) +Q 𝑄))
37 vex 2733 . . . . . . 7 𝑥 ∈ V
38 breq1 3992 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹𝐾) +Q 𝑄)))
3937, 38elab 2874 . . . . . 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 7511 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)} ∈ V
42 gtnqex 7512 . . . . . 6 {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢} ∈ V
4341, 42op1st 6125 . . . . 5 (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩) = {𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}
4440, 43eleqtrrdi 2264 . . . 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 2519 . . . 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 1231 . . 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 7638 . . . . 5 (𝜑𝐿P)
5049adantr 274 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥𝑥 <Q ((𝐹𝐾) +Q 𝑄)))) → 𝐿P)
51 caucvgprlemlim.q . . . . . . 7 (𝜑𝑄Q)
52 addclnq 7337 . . . . . . 7 (((𝐹𝐾) ∈ Q𝑄Q) → ((𝐹𝐾) +Q 𝑄) ∈ Q)
539, 51, 52syl2anc 409 . . . . . 6 (𝜑 → ((𝐹𝐾) +Q 𝑄) ∈ Q)
54 nqprlu 7509 . . . . . 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 274 . . . 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 7468 . . . 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 409 . . 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 2592 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 1348  wcel 2141  {cab 2156  wral 2448  wrex 2449  {crab 2452  cop 3586   class class class wbr 3989  wf 5194  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  1oc1o 6388  [cec 6511  Ncnpi 7234   <N clti 7237   ~Q ceq 7241  Qcnq 7242   +Q cplq 7244  *Qcrq 7246   <Q cltq 7247  Pcnp 7253  <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432
This theorem is referenced by:  caucvgprlemlim  7643
  Copyright terms: Public domain W3C validator