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Theorem caucvgprprlem1 6805
Description: Lemma for caucvgprpr 6808. 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 6420 . . . . . 6 <N ⊆ (N × N)
76brel 4392 . . . . 5 (𝐽 <N 𝐾 → (𝐽N𝐾N))
85, 7syl 14 . . . 4 (𝜑 → (𝐽N𝐾N))
98simprd 107 . . 3 (𝜑𝐾N)
101, 9ffvelrnd 5303 . 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 6779 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
14 nnnq 6518 . . . . . . . 8 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
159, 14syl 14 . . . . . . 7 (𝜑 → [⟨𝐾, 1𝑜⟩] ~QQ)
16 recclnq 6488 . . . . . . 7 ([⟨𝐾, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
17 nqprlu 6643 . . . . . . 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 6715 . . . . . 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 1135 . . . . 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 135 . . . 4 (𝜑 → ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝐾) +P 𝑄))
22 opeq1 3549 . . . . . . . . . . . 12 (𝑟 = 𝐾 → ⟨𝑟, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
2322eceq1d 6142 . . . . . . . . . . 11 (𝑟 = 𝐾 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
2423fveq2d 5182 . . . . . . . . . 10 (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
2524breq2d 3776 . . . . . . . . 9 (𝑟 = 𝐾 → (𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
2625abbidv 2155 . . . . . . . 8 (𝑟 = 𝐾 → {𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )})
2724breq1d 3774 . . . . . . . . 9 (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢))
2827abbidv 2155 . . . . . . . 8 (𝑟 = 𝐾 → {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢})
2926, 28opeq12d 3557 . . . . . . 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 5528 . . . . . 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 5178 . . . . . . 7 (𝑟 = 𝐾 → (𝐹𝑟) = (𝐹𝐾))
3231oveq1d 5527 . . . . . 6 (𝑟 = 𝐾 → ((𝐹𝑟) +P 𝑄) = ((𝐹𝐾) +P 𝑄))
3330, 32breq12d 3777 . . . . 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 2656 . . . 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 391 . . 3 (𝜑 → ∃𝑟N ((𝐹𝐾) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹𝑟) +P 𝑄))
36 breq1 3767 . . . . . . . 8 (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
3736cbvabv 2161 . . . . . . 7 {𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}
38 breq2 3768 . . . . . . . 8 (𝑢 = 𝑞 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞))
3938cbvabv 2161 . . . . . . 7 {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}
4037, 39opeq12i 3554 . . . . . 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 5523 . . . . 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 3771 . . . 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 2331 . . 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 127 . 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 6804 1 (𝜑 → (𝐹𝐾)<P (𝐿 +P 𝑄))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  {crab 2310  cop 3378   class class class wbr 3764  wf 4898  cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  Qcnq 6376   +Q cplq 6378  *Qcrq 6380   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  caucvgprprlemlim  6807
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