Theorem caucvgprprlemlim 6809

Description: Lemma for caucvgprpr 6810. 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   𝑟,𝑙,𝑞   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 6762	. . . 4 ⊢ (𝑥 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑥)
2	1	adantl 262	. . 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 465	. . . . . . . . 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 465	. . . . . . . . 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 465	. . . . . . . . 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 103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ P) → 𝑥 ∈ P)
11	10	ad4antr 463	. . . . . . . . 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 103	. . . . . . . . 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 486	. . . . . . . . 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 6807	. . . . . . . 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 6808	. . . . . . . 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 290	. . . . . . 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 108	. . . . . 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 2392	. . . . 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 108	. . . 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 2421	. . 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 2392	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 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389   +_P cpp 6391  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568

This theorem is referenced by:  caucvgprpr  6810