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Theorem climcl 15208
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 15201 . . . . 5 Rel ⇝
21brrelex1i 5643 . . . 4 (𝐹𝐴𝐹 ∈ V)
3 eqidd 2739 . . . 4 ((𝐹𝐴𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
42, 3clim 15203 . . 3 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
54ibi 266 . 2 (𝐹𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
65simpld 495 1 (𝐹𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3064  wrex 3065  Vcvv 3432   class class class wbr 5074  cfv 6433  (class class class)co 7275  cc 10869   < clt 11009  cmin 11205  cz 12319  cuz 12582  +crp 12730  abscabs 14945  cli 15193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-cnex 10927  ax-resscn 10928
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-neg 11208  df-z 12320  df-uz 12583  df-clim 15197
This theorem is referenced by:  rlimclim  15255  climrlim2  15256  climuni  15261  fclim  15262  climeu  15264  climreu  15265  2clim  15281  climcn1lem  15312  climadd  15341  climmul  15342  climsub  15343  climaddc2  15345  climcau  15382  clim2div  15601  ntrivcvgtail  15612  ntrivcvgmullem  15613  mbflim  24832  ulmcau  25554  emcllem6  26150  dchrmusum2  26642  dchrvmasumiflem1  26649  dchrvmasumiflem2  26650  dchrisum0lem1b  26663  dchrmusumlem  26670  iprodefisum  33707  climrec  43144  climexp  43146  climsuse  43149  climneg  43151  climdivf  43153  climleltrp  43217  climuzlem  43284  climxlim2lem  43386  climxlim2  43387  sge0isum  43965
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