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Theorem caucvgprlem1 7680
Description: Lemma for caucvgpr 7683. 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 7325 . . . . . . 7 <N ⊆ (N × N)
32brel 4680 . . . . . 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 7666 . . . . 5 (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
8 caucvgpr.f . . . . . 6 (𝜑𝐹:NQ)
98, 5ffvelcdmd 5654 . . . . 5 (𝜑 → (𝐹𝐾) ∈ Q)
10 ltanqi 7403 . . . . 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 3780 . . . . . . . . 9 (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
1312eceq1d 6573 . . . . . . . 8 (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
1413fveq2d 5521 . . . . . . 7 (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
1514oveq2d 5893 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
16 fveq2 5517 . . . . . . 7 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
1716oveq1d 5892 . . . . . 6 (𝑗 = 𝐾 → ((𝐹𝑗) +Q 𝑄) = ((𝐹𝐾) +Q 𝑄))
1815, 17breq12d 4018 . . . . 5 (𝑗 = 𝐾 → (((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄) ↔ ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹𝐾) +Q 𝑄)))
1918rspcev 2843 . . . 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 5884 . . . . . . . 8 (𝑙 = (𝐹𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2221breq1d 4015 . . . . . . 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 2896 . . . . 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 7679 . . . . 5 (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑄)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
3130sseld 3156 . . . 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 7677 . . . 4 (𝜑𝐿P)
35 nqprlu 7548 . . . . 5 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
3629, 35syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
37 addclpr 7538 . . . 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 7552 . . 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 3597   class class class wbr 4005  wf 5214  cfv 5218  (class class class)co 5877  1st c1st 6141  1oc1o 6412  [cec 6535  Ncnpi 7273   <N clti 7276   ~Q ceq 7280  Qcnq 7281   +Q cplq 7283  *Qcrq 7285   <Q cltq 7286  Pcnp 7292   +P cpp 7294  <P cltp 7296
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471
This theorem is referenced by:  caucvgprlemlim  7682
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