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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
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