Theorem caucvgprlemlim 7622

Description: Lemma for caucvgpr 7623. 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 7604	. . . 4 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
2	1	adantl 275	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
3		caucvgpr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:N⟶Q)
4	3	ad5antr 488	. . . . . . . . 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 488	. . . . . . . . 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 488	. . . . . . . . 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 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Q) → 𝑥 ∈ Q)
11	10	ad4antr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝑥 ∈ Q)
12		simpr 109	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ Q) ∧ 𝑗 ∈ N) ∧ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) ∧ 𝑘 ∈ N) ∧ 𝑗 <N 𝑘) → 𝑗 <N 𝑘)
13		simpllr 524	. . . . . . . . 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 7620	. . . . . . . 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 7621	. . . . . . . 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 304	. . . . . . 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 114	. . . . . 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 2539	. . . . 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 114	. . . 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 2568	. . 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 2539	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 103   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  {crab 2448  ⟨cop 3579   class class class wbr 3982  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1_oc1o 6377  [cec 6499  Ncnpi 7213   <_N clti 7216   ~_Q ceq 7220  Qcnq 7221   +_Q cplq 7223  *_Qcrq 7225   <_Q cltq 7226   +_P cpp 7234  <_P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by:  caucvgpr  7623