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Theorem climi 14235
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
climi (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (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 climi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 climi.3 . 2 (𝜑𝐶 ∈ ℝ+)
2 climi.5 . . . 4 (𝜑𝐹𝐴)
3 climi.1 . . . . 5 𝑍 = (ℤ𝑀)
4 climi.2 . . . . 5 (𝜑𝑀 ∈ ℤ)
5 climrel 14217 . . . . . . 7 Rel ⇝
65brrelexi 5156 . . . . . 6 (𝐹𝐴𝐹 ∈ V)
72, 6syl 17 . . . . 5 (𝜑𝐹 ∈ V)
8 climi.4 . . . . 5 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
93, 4, 7, 8clim2 14229 . . . 4 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
102, 9mpbid 222 . . 3 (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1110simprd 479 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))
12 breq2 4655 . . . . 5 (𝑥 = 𝐶 → ((abs‘(𝐵𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝐶))
1312anbi2d 740 . . . 4 (𝑥 = 𝐶 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶)))
1413rexralbidv 3056 . . 3 (𝑥 = 𝐶 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶)))
1514rspcv 3303 . 2 (𝐶 ∈ ℝ+ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶)))
161, 11, 15sylc 65 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912  Vcvv 3198   class class class wbr 4651  cfv 5886  (class class class)co 6647  cc 9931   < clt 10071  cmin 10263  cz 11374  cuz 11684  +crp 11829  abscabs 13968  cli 14209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-pre-lttri 10007  ax-pre-lttrn 10008
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-po 5033  df-so 5034  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-neg 10266  df-z 11375  df-uz 11685  df-clim 14213
This theorem is referenced by:  climi2  14236  climi0  14237  climuni  14277  2clim  14297  climcau  14395  caucvgb  14404  stoweidlem7  39993
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