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Theorem caucvgprlem1 6931
 Description: Lemma for caucvgpr 6934. 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
caucvgprlem1 (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝐾)}, {𝑢 ∣ (𝐹𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙,𝑢   𝑗,𝐾,𝑙,𝑢   𝑄,𝑗,𝑙,𝑢   𝑄,𝑘   𝑗,𝐿,𝑘   𝑢,𝑗   𝑘,𝐹,𝑛   𝑗,𝑘
Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝑄(𝑛)   𝐽(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐾(𝑘,𝑛)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlem1
StepHypRef Expression
1 caucvgprlemlim.jk . . . . . 6 (𝜑𝐽 <N 𝐾)
2 ltrelpi 6576 . . . . . . 7 <N ⊆ (N × N)
32brel 4418 . . . . . 6 (𝐽 <N 𝐾 → (𝐽N𝐾N))
41, 3syl 14 . . . . 5 (𝜑 → (𝐽N𝐾N))
54simprd 112 . . . 4 (𝜑𝐾N)
6 caucvgprlemlim.jkq . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)
71, 6caucvgprlemk 6917 . . . . 5 (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄)
8 caucvgpr.f . . . . . 6 (𝜑𝐹:NQ)
98, 5ffvelrnd 5335 . . . . 5 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 6654 . . . . 5 (((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
117, 9, 10syl2anc 403 . . . 4 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 opeq1 3578 . . . . . . . . 9 (𝑗 = 𝐾 → ⟨𝑗, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
1312eceq1d 6208 . . . . . . . 8 (𝑗 = 𝐾 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
1413fveq2d 5213 . . . . . . 7 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
1514oveq2d 5559 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
16 fveq2 5209 . . . . . . 7 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
1716oveq1d 5558 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝑗) +Q 𝑄) = ((𝐹𝐾) +Q 𝑄))
1815, 17breq12d 3806 . . . . 5 (𝑗 = 𝐾 → (((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄)))
1918rspcev 2702 . . . 4 ((𝐾N ∧ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄)) → ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄))
205, 11, 19syl2anc 403 . . 3 (𝜑 → ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄))
21 oveq1 5550 . . . . . . . 8 (𝑙 = (𝐹𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
2221breq1d 3803 . . . . . . 7 (𝑙 = (𝐹𝐾) → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)))
2322rexbidv 2370 . . . . . 6 (𝑙 = (𝐹𝐾) → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)))
2423elrab3 2751 . . . . 5 ((𝐹𝐾) ∈ Q → ((𝐹𝐾) ∈ {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} ↔ ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)))
259, 24syl 14 . . . 4 (𝜑 → ((𝐹𝐾) ∈ {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} ↔ ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)))
26 caucvgpr.cau . . . . . 6 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
27 caucvgpr.bnd . . . . . 6 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
28 caucvgpr.lim . . . . . 6 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
29 caucvgprlemlim.q . . . . . 6 (𝜑𝑄Q)
308, 26, 27, 28, 29caucvgprlemladdrl 6930 . . . . 5 (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
3130sseld 2999 . . . 4 (𝜑 → ((𝐹𝐾) ∈ {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} → (𝐹𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))))
3225, 31sylbird 168 . . 3 (𝜑 → (∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) → (𝐹𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))))
3320, 32mpd 13 . 2 (𝜑 → (𝐹𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
348, 26, 27, 28caucvgprlemcl 6928 . . . 4 (𝜑𝐿P)
35 nqprlu 6799 . . . . 5 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
3629, 35syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
37 addclpr 6789 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
3834, 36, 37syl2anc 403 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
39 nqprl 6803 . . 3 (((𝐹𝐾) ∈ Q ∧ (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P) → ((𝐹𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝐹𝐾)}, {𝑢 ∣ (𝐹𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
409, 38, 39syl2anc 403 . 2 (𝜑 → ((𝐹𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝐹𝐾)}, {𝑢 ∣ (𝐹𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
4133, 40mpbid 145 1 (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝐾)}, {𝑢 ∣ (𝐹𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285   ∈ wcel 1434  {cab 2068  ∀wral 2349  ∃wrex 2350  {crab 2353  ⟨cop 3409   class class class wbr 3793  ⟶wf 4928  ‘cfv 4932  (class class class)co 5543  1st c1st 5796  1𝑜c1o 6058  [cec 6170  Ncnpi 6524
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