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Theorem climcl 14274
Description: Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
climcl (𝐹𝐴𝐴 ∈ ℂ)

Proof of Theorem climcl
Dummy variables 𝑥 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climrel 14267 . . . . 5 Rel ⇝
21brrelexi 5192 . . . 4 (𝐹𝐴𝐹 ∈ V)
3 eqidd 2652 . . . 4 ((𝐹𝐴𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
42, 3clim 14269 . . 3 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
54ibi 256 . 2 (𝐹𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
65simpld 474 1 (𝐹𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cfv 5926  (class class class)co 6690  cc 9972   < clt 10112  cmin 10304  cz 11415  cuz 11725  +crp 11870  abscabs 14018  cli 14259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-neg 10307  df-z 11416  df-uz 11726  df-clim 14263
This theorem is referenced by:  rlimclim  14321  climrlim2  14322  climuni  14327  fclim  14328  climeu  14330  climreu  14331  2clim  14347  climcn1lem  14377  climadd  14406  climmul  14407  climsub  14408  climaddc2  14410  climcau  14445  clim2div  14665  ntrivcvgtail  14676  ntrivcvgmullem  14677  mbflim  23480  ulmcau  24194  emcllem6  24772  dchrmusum2  25228  dchrvmasumiflem1  25235  dchrvmasumiflem2  25236  dchrisum0lem1b  25249  dchrmusumlem  25256  iprodefisum  31753  climrec  40153  climexp  40155  climsuse  40158  climneg  40160  climdivf  40162  climleltrp  40226  climuzlem  40293  climxlim2lem  40389  climxlim2  40390  sge0isum  40962
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