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Theorem caucvgprlemopu 7151
Description: Lemma for caucvgpr 7162. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemopu ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
Distinct variable groups:   𝐴,𝑗   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹   𝑗,𝐿,𝑟,𝑠   𝑗,𝑙,𝑠   𝜑,𝑗,𝑟,𝑠   𝑢,𝑗,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑗,𝑘,𝑛)   𝐿(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprlemopu
StepHypRef Expression
1 breq2 3818 . . . . . 6 (𝑢 = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
21rexbidv 2377 . . . . 5 (𝑢 = 𝑟 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
3 caucvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
43fveq2i 5259 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
5 nqex 6843 . . . . . . . 8 Q ∈ V
65rabex 3951 . . . . . . 7 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
75rabex 3951 . . . . . . 7 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
86, 7op2nd 5856 . . . . . 6 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
94, 8eqtri 2105 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
102, 9elrab2 2764 . . . 4 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
1110simprbi 269 . . 3 (𝑟 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
1211adantl 271 . 2 ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
13 simprr 499 . . . 4 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
14 ltbtwnnqq 6895 . . . 4 (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟 ↔ ∃𝑠Q (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟))
1513, 14sylib 120 . . 3 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → ∃𝑠Q (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟))
16 simprr 499 . . . . . 6 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑠 <Q 𝑟)
17 simplr 497 . . . . . . 7 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑠Q)
18 simplrl 502 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) → 𝑗N)
1918adantr 270 . . . . . . . 8 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑗N)
20 simprl 498 . . . . . . . 8 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
21 rspe 2420 . . . . . . . 8 ((𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
2219, 20, 21syl2anc 403 . . . . . . 7 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
23 breq2 3818 . . . . . . . . 9 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2423rexbidv 2377 . . . . . . . 8 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2524, 9elrab2 2764 . . . . . . 7 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2617, 22, 25sylanbrc 408 . . . . . 6 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑠 ∈ (2nd𝐿))
2716, 26jca 300 . . . . 5 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
2827ex 113 . . . 4 ((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟) → (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
2928reximdva 2471 . . 3 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → (∃𝑠Q (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
3015, 29mpd 13 . 2 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
3112, 30rexlimddv 2489 1 ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  wral 2355  wrex 2356  {crab 2359  cop 3428   class class class wbr 3814  wf 4968  cfv 4972  (class class class)co 5594  2nd c2nd 5848  1𝑜c1o 6109  [cec 6223  Ncnpi 6752   <N clti 6755   ~Q ceq 6759  Qcnq 6760   +Q cplq 6762  *Qcrq 6764   <Q cltq 6765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833
This theorem is referenced by:  caucvgprlemrnd  7153
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