Theorem caucvgprprlemlim 7712

Description: Lemma for caucvgprpr 7713. 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 7665	. . . 4 ⊢ (𝑥 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
2	1	adantl 277	. . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ P) → 𝑥 ∈ P)
11	10	ad4antr 494	. . . . . . . . 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 110	. . . . . . . . 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 534	. . . . . . . . 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 7710	. . . . . . . 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 7711	. . . . . . . 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 306	. . . . . . 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 115	. . . . . 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 2550	. . . . 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 115	. . . 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 2579	. . 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 2550	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 104   = 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  1_oc1o 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:  caucvgprpr  7713