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Theorem caucvgprprlem1 6961
Description: Lemma for caucvgprpr 6964. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
caucvgprprlemlim.q (𝜑𝑄P)
caucvgprprlemlim.jk (𝜑𝐽 <N 𝐾)
caucvgprprlemlim.jkq (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
Assertion
Ref Expression
caucvgprprlem1 (𝜑 → (𝐹𝐾)<P (𝐿 +P 𝑄))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟   𝐹,𝑟,𝑙,𝑢,𝑛,𝑘   𝐽,𝑙,𝑢   𝐾,𝑙,𝑟,𝑢   𝑄,𝑟   𝑘,𝐿   𝜑,𝑟   𝑞,𝑝,𝑟,𝑙,𝑢   𝑚,𝑟   𝑘,𝑙,𝑢,𝑟,𝑝,𝑞   𝑛,𝑙,𝑢,𝑟
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑄(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑞,𝑝)   𝐽(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝)   𝐾(𝑘,𝑚,𝑛,𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlem1
StepHypRef Expression
1 caucvgprpr.f . 2 (𝜑𝐹:NP)
2 caucvgprpr.cau . 2 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
3 caucvgprpr.bnd . 2 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 caucvgprpr.lim . 2 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
5 caucvgprprlemlim.jk . . . . 5 (𝜑𝐽 <N 𝐾)
6 ltrelpi 6576 . . . . . 6 <N ⊆ (N × N)
76brel 4418 . . . . 5 (𝐽 <N 𝐾 → (𝐽N𝐾N))
85, 7syl 14 . . . 4 (𝜑 → (𝐽N𝐾N))
98simprd 112 . . 3 (𝜑𝐾N)
101, 9ffvelrnd 5335 . 2 (𝜑 → (𝐹𝐾) ∈ P)
11 caucvgprprlemlim.q . 2 (𝜑𝑄P)
12 caucvgprprlemlim.jkq . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
135, 12caucvgprprlemk 6935 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
14 nnnq 6674 . . . . . . . 8 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
159, 14syl 14 . . . . . . 7 (𝜑 → [⟨𝐾, 1𝑜⟩] ~QQ)
16 recclnq 6644 . . . . . . 7 ([⟨𝐾, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
17 nqprlu 6799 . . . . . . 7 ((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
1815, 16, 173syl 17 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
19 ltaprg 6871 . . . . . 6 ((⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P𝑄P ∧ (𝐹𝐾) ∈ P) → (⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄)))
2018, 11, 10, 19syl3anc 1170 . . . . 5 (𝜑 → (⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄)))
2113, 20mpbid 145 . . . 4 (𝜑 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄))
22 opeq1 3578 . . . . . . . . . . . 12 (𝑟 = 𝐾 → ⟨𝑟, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
2322eceq1d 6208 . . . . . . . . . . 11 (𝑟 = 𝐾 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
2423fveq2d 5213 . . . . . . . . . 10 (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
2524breq2d 3805 . . . . . . . . 9 (𝑟 = 𝐾 → (𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
2625abbidv 2197 . . . . . . . 8 (𝑟 = 𝐾 → {𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )})
2724breq1d 3803 . . . . . . . . 9 (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢))
2827abbidv 2197 . . . . . . . 8 (𝑟 = 𝐾 → {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢})
2926, 28opeq12d 3586 . . . . . . 7 (𝑟 = 𝐾 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
3029oveq2d 5559 . . . . . 6 (𝑟 = 𝐾 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
31 fveq2 5209 . . . . . . 7 (𝑟 = 𝐾 → (𝐹𝑟) = (𝐹𝐾))
3231oveq1d 5558 . . . . . 6 (𝑟 = 𝐾 → ((𝐹𝑟) +P 𝑄) = ((𝐹𝐾) +P 𝑄))
3330, 32breq12d 3806 . . . . 5 (𝑟 = 𝐾 → (((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝑟) +P 𝑄) ↔ ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄)))
3433rspcev 2702 . . . 4 ((𝐾N ∧ ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄)) → ∃𝑟N ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝑟) +P 𝑄))
359, 21, 34syl2anc 403 . . 3 (𝜑 → ∃𝑟N ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝑟) +P 𝑄))
36 breq1 3796 . . . . . . . 8 (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
3736cbvabv 2203 . . . . . . 7 {𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}
38 breq2 3797 . . . . . . . 8 (𝑢 = 𝑞 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞))
3938cbvabv 2203 . . . . . . 7 {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}
4037, 39opeq12i 3583 . . . . . 6 ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩
4140oveq2i 5554 . . . . 5 ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)
4241breq1i 3800 . . . 4 (((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝑟) +P 𝑄) ↔ ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
4342rexbii 2374 . . 3 (∃𝑟N ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝑟) +P 𝑄) ↔ ∃𝑟N ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
4435, 43sylib 120 . 2 (𝜑 → ∃𝑟N ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
451, 2, 3, 4, 10, 11, 44caucvgprprlemaddq 6960 1 (𝜑 → (𝐹𝐾)<P (𝐿 +P 𝑄))
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  1𝑜c1o 6058  [cec 6170  Ncnpi 6524   <N clti 6527   ~Q ceq 6531  Qcnq 6532   +Q cplq 6534  *Qcrq 6536   <Q cltq 6537  Pcnp 6543   +P cpp 6545  <P cltp 6547
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprprlemlim  6963
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