Theorem caucvgprlemlim 8042

Description: Lemma for caucvgpr 8043. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩

Assertion

Ref	Expression
caucvgprlemlim	⊢ (𝜑 → ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑢,𝑙,𝑘   𝑛,𝐹,𝑘   𝑗,𝑘,𝜑,𝑥   𝑘,𝑙,𝑢,𝑥,𝑗   𝑗,𝐿,𝑘

Allowed substitution hints:   𝜑(𝑢,𝑛,𝑙)   𝐴(𝑥,𝑢,𝑘,𝑛,𝑙)   𝐹(𝑥)   𝐿(𝑥,𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlemlim

Step	Hyp	Ref	Expression
1		archrecnq 8024	. . . 4 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
2	1	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
3		caucvgpr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:N⟶Q)
4	3	ad5antr 500	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝐹:N⟶Q)
5		caucvgpr.cau	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
6	5	ad5antr 500	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
7		caucvgpr.bnd	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
8	7	ad5antr 500	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
9		caucvgpr.lim	. . . . . . . . 9 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
10		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → 𝑥 ∈ Q)
11	10	ad4antr 498	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝑥 ∈ Q)
12		simpr 110	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝑗 <N 𝑘)
13		simpllr 540	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
14	4, 6, 8, 9, 11, 12, 13	caucvgprlem1 8040	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩))
15	4, 6, 8, 9, 11, 12, 13	caucvgprlem2 8041	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)
16	14, 15	jca 306	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))
17	16	ex 115	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) → (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
18	17	ralrimiva 2623	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) → ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
19	18	ex 115	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))))
20	19	reximdva 2652	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → (∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))))
21	2, 20	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
22	21	ralrimiva 2623	1 ⊢ (𝜑 → ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3711   class class class wbr 4128  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079  1_oc1o 6674  [cec 6799  Ncnpi 7633   <_N clti 7636   ~_Q ceq 7640  Qcnq 7641   +_Q cplq 7643  *_Qcrq 7645   <_Q cltq 7646   +_P cpp 7654  <_P cltp 7656

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-iplp 7829  df-iltp 7831

This theorem is referenced by:  caucvgpr  8043