Theorem caucvgprprlemlim 7461

Description: Lemma for caucvgprpr 7462. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
caucvgprprlemlim	⊢ (𝜑 → ∀𝑥 ∈ P ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥))))

Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑗   𝑢,𝐹,𝑟,𝑙,𝑘,𝑛   𝜑,𝑘,𝑟   𝑘,𝐿   𝑗,𝑘,𝜑,𝑥   𝑘,𝑙,𝑢,𝑝,𝑞,𝑟   𝑗,𝑟,𝑥   𝑞,𝑙,𝑟   𝑢,𝑝,𝑞,𝑟   𝑚,𝑟   𝑘,𝑛,𝑢,𝑙   𝑗,𝑙,𝑢   𝑛,𝑟

Allowed substitution hints:   𝜑(𝑢,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑥,𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑥,𝑗,𝑞,𝑝)   𝐿(𝑥,𝑢,𝑗,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemlim

Step	Hyp	Ref	Expression
1		archrecpr 7414	. . . 4 ⊢ (𝑥 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
2	1	adantl 273	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ P) → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
3		caucvgprpr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:N⟶P)
4	3	ad5antr 485	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝐹:N⟶P)
5		caucvgprpr.cau	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
6	5	ad5antr 485	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
7		caucvgprpr.bnd	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
8	7	ad5antr 485	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
9		caucvgprpr.lim	. . . . . . . . 9 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
10		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ P) → 𝑥 ∈ P)
11	10	ad4antr 483	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝑥 ∈ P)
12		simpr 109	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝑗 <N 𝑘)
13		simpllr 506	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
14	4, 6, 8, 9, 11, 12, 13	caucvgprprlem1 7459	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → (𝐹‘𝑘)<P (𝐿 +P 𝑥))
15	4, 6, 8, 9, 11, 12, 13	caucvgprprlem2 7460	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝐿<P ((𝐹‘𝑘) +P 𝑥))
16	14, 15	jca 302	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥)))
17	16	ex 114	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) ∧ 𝑘 ∈ N) → (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥))))
18	17	ralrimiva 2477	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥) → ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥))))
19	18	ex 114	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ P) ∧ 𝑗 ∈ N) → (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥 → ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥)))))
20	19	reximdva 2506	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ P) → (∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥)))))
21	2, 20	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ P) → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥))))
22	21	ralrimiva 2477	1 ⊢ (𝜑 → ∀𝑥 ∈ P ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥))))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1312   ∈ wcel 1461  {cab 2099  ∀wral 2388  ∃wrex 2389  {crab 2392  ⟨cop 3494   class class class wbr 3893  ⟶wf 5075  ‘cfv 5079  (class class class)co 5726  1_oc1o 6258  [cec 6379  Ncnpi 7022   <_N clti 7025   ~_Q ceq 7029  Qcnq 7030   +_Q cplq 7032  *_Qcrq 7034   <_Q cltq 7035  Pcnp 7041   +_P cpp 7043  <_P cltp 7045

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-iplp 7218  df-iltp 7220

This theorem is referenced by:  caucvgprpr  7462