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Theorem caucvgprlem1 7677
Description: Lemma for caucvgpr 7680. 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 7322 . . . . . . 7 <N ⊆ (N × N)
32brel 4678 . . . . . 6 (𝐽 <N 𝐾 → (𝐽N𝐾N))
41, 3syl 14 . . . . 5 (𝜑 → (𝐽N𝐾N))
54simprd 114 . . . 4 (𝜑𝐾N)
6 caucvgprlemlim.jkq . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)
71, 6caucvgprlemk 7663 . . . . 5 (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
8 caucvgpr.f . . . . . 6 (𝜑𝐹:NQ)
98, 5ffvelcdmd 5652 . . . . 5 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 7400 . . . . 5 (((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄 ∧ (𝐹𝐾) ∈ Q) → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
117, 9, 10syl2anc 411 . . . 4 (𝜑 → ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄))
12 opeq1 3778 . . . . . . . . 9 (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
1312eceq1d 6570 . . . . . . . 8 (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
1413fveq2d 5519 . . . . . . 7 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
1514oveq2d 5890 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
16 fveq2 5515 . . . . . . 7 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
1716oveq1d 5889 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝑗) +Q 𝑄) = ((𝐹𝐾) +Q 𝑄))
1815, 17breq12d 4016 . . . . 5 (𝑗 = 𝐾 → (((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄)))
1918rspcev 2841 . . . 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 411 . . 3 (𝜑 → ∃𝑗N ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄))
21 oveq1 5881 . . . . . . . 8 (𝑙 = (𝐹𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2221breq1d 4013 . . . . . . 7 (𝑙 = (𝐹𝐾) → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)))
2322rexbidv 2478 . . . . . 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 2894 . . . . 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 7676 . . . . 5 (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
3130sseld 3154 . . . 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 170 . . 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 7674 . . . 4 (𝜑𝐿P)
35 nqprlu 7545 . . . . 5 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
3629, 35syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
37 addclpr 7535 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
3834, 36, 37syl2anc 411 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
39 nqprl 7549 . . 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 411 . 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 147 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 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  cop 3595   class class class wbr 4003  wf 5212  cfv 5216  (class class class)co 5874  1st c1st 6138  1oc1o 6409  [cec 6532  Ncnpi 7270   <N clti 7273   ~Q ceq 7277  Qcnq 7278   +Q cplq 7280  *Qcrq 7282   <Q cltq 7283  Pcnp 7289   +P cpp 7291  <P cltp 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468
This theorem is referenced by:  caucvgprlemlim  7679
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