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Theorem climi 14618
 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 breq2 4877 . . . 4 (𝑥 = 𝐶 → ((abs‘(𝐵𝐴)) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝐶))
21anbi2d 624 . . 3 (𝑥 = 𝐶 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶)))
32rexralbidv 3268 . 2 (𝑥 = 𝐶 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶)))
4 climi.5 . . . 4 (𝜑𝐹𝐴)
5 climi.1 . . . . 5 𝑍 = (ℤ𝑀)
6 climi.2 . . . . 5 (𝜑𝑀 ∈ ℤ)
7 climrel 14600 . . . . . . 7 Rel ⇝
87brrelex1i 5393 . . . . . 6 (𝐹𝐴𝐹 ∈ V)
94, 8syl 17 . . . . 5 (𝜑𝐹 ∈ V)
10 climi.4 . . . . 5 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)
115, 6, 9, 10clim2 14612 . . . 4 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
124, 11mpbid 224 . . 3 (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥)))
1312simprd 491 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))
14 climi.3 . 2 (𝜑𝐶 ∈ ℝ+)
153, 13, 14rspcdva 3532 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ∃wrex 3118  Vcvv 3414   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  ℂcc 10250   < clt 10391   − cmin 10585  ℤcz 11704  ℤ≥cuz 11968  ℝ+crp 12112  abscabs 14351   ⇝ cli 14592 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-pre-lttri 10326  ax-pre-lttrn 10327 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-po 5263  df-so 5264  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-neg 10588  df-z 11705  df-uz 11969  df-clim 14596 This theorem is referenced by:  climi2  14619  climi0  14620  climuni  14660  2clim  14680  climcau  14778  caucvgb  14787  stoweidlem7  41018
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