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Theorem caucvgprlem1 6805
Description: Lemma for caucvgpr 6808. 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 6450 . . . . . . 7 <N ⊆ (N × N)
32brel 4417 . . . . . 6 (𝐽 <N 𝐾 → (𝐽N𝐾N))
41, 3syl 14 . . . . 5 (𝜑 → (𝐽N𝐾N))
54simprd 111 . . . 4 (𝜑𝐾N)
6 caucvgprlemlim.jkq . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)
71, 6caucvgprlemk 6791 . . . . 5 (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄)
8 caucvgpr.f . . . . . 6 (𝜑𝐹:NQ)
98, 5ffvelrnd 5328 . . . . 5 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 6528 . . . . 5 (((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
117, 9, 10syl2anc 397 . . . 4 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 opeq1 3574 . . . . . . . . 9 (𝑗 = 𝐾 → ⟨𝑗, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
1312eceq1d 6170 . . . . . . . 8 (𝑗 = 𝐾 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
1413fveq2d 5207 . . . . . . 7 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
1514oveq2d 5553 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
16 fveq2 5203 . . . . . . 7 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
1716oveq1d 5552 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝑗) +Q 𝑄) = ((𝐹𝐾) +Q 𝑄))
1815, 17breq12d 3802 . . . . 5 (𝑗 = 𝐾 → (((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄)))
1918rspcev 2671 . . . 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 397 . . 3 (𝜑 → ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄))
21 oveq1 5544 . . . . . . . 8 (𝑙 = (𝐹𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
2221breq1d 3799 . . . . . . 7 (𝑙 = (𝐹𝐾) → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)))
2322rexbidv 2342 . . . . . 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 2719 . . . . 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 6804 . . . . 5 (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
3130sseld 2969 . . . 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 163 . . 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 6802 . . . 4 (𝜑𝐿P)
35 nqprlu 6673 . . . . 5 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
3629, 35syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
37 addclpr 6663 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
3834, 36, 37syl2anc 397 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
39 nqprl 6677 . . 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 397 . 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 139 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 101  wb 102   = wceq 1257  wcel 1407  {cab 2040  wral 2321  wrex 2322  {crab 2325  cop 3403   class class class wbr 3789  wf 4923  cfv 4927  (class class class)co 5537  1st c1st 5790  1𝑜c1o 6022  [cec 6132  Ncnpi 6398   <N clti 6401   ~Q ceq 6405  Qcnq 6406   +Q cplq 6408  *Qcrq 6410   <Q cltq 6411  Pcnp 6417   +P cpp 6419  <P cltp 6421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-2o 6030  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-enq0 6550  df-nq0 6551  df-0nq0 6552  df-plq0 6553  df-mq0 6554  df-inp 6592  df-iplp 6594  df-iltp 6596
This theorem is referenced by:  caucvgprlemlim  6807
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