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Theorem climi2 14867
Description: Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climi.1 𝑍 = (ℤ𝑀)
climi.2 (𝜑𝑀 ∈ ℤ)
climi.3 (𝜑𝐶 ∈ ℝ+)
climi.4 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
climi.5 (𝜑𝐹𝐴)
Assertion
Ref Expression
climi2 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)
Distinct variable groups:   𝑗,𝑘,𝐴   𝐶,𝑗,𝑘   𝑗,𝐹,𝑘   𝜑,𝑗,𝑘   𝑗,𝑍,𝑘   𝑗,𝑀
Allowed substitution hints:   𝐵(𝑗,𝑘)   𝑀(𝑘)

Proof of Theorem climi2
StepHypRef Expression
1 climi.1 . . 3 𝑍 = (ℤ𝑀)
2 climi.2 . . 3 (𝜑𝑀 ∈ ℤ)
3 climi.3 . . 3 (𝜑𝐶 ∈ ℝ+)
4 climi.4 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
5 climi.5 . . 3 (𝜑𝐹𝐴)
61, 2, 3, 4, 5climi 14866 . 2 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶))
7 simpr 487 . . . 4 ((𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶) → (abs‘(𝐵𝐴)) < 𝐶)
87ralimi 3160 . . 3 (∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶) → ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)
98reximi 3243 . 2 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)
106, 9syl 17 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  wral 3138  wrex 3139   class class class wbr 5065  cfv 6354  (class class class)co 7155  cc 10534   < clt 10674  cmin 10869  cz 11980  cuz 12242  +crp 12388  abscabs 14592  cli 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-pre-lttri 10610  ax-pre-lttrn 10611
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-neg 10872  df-z 11981  df-uz 12243  df-clim 14844
This theorem is referenced by:  rlimclim  14902  climcn1  14947  climcn2  14948  climsqz  14996  climsqz2  14997  mertenslem2  15240  uniioombllem6  24188  ulmcau  24982  ulmdvlem3  24989  rrncmslem  35109  cvgdvgrat  40645  stoweidlem7  42293
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