Theorem caucvgprlemlim 7944

Description: Lemma for caucvgpr 7945. 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 7926	. . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 494	. . . . . . . . 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 536	. . . . . . . . 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 7942	. . . . . . . 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 7943	. . . . . . . 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 2606	. . . . 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 2635	. . 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 2606	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 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1_oc1o 6618  [cec 6743  Ncnpi 7535   <_N clti 7538   ~_Q ceq 7542  Qcnq 7543   +_Q cplq 7545  *_Qcrq 7547   <_Q cltq 7548   +_P cpp 7556  <_P cltp 7558

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-iltp 7733

This theorem is referenced by:  caucvgpr  7945