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Theorem climcl 14939
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 14932 . . . . 5 Rel ⇝
21brrelex1i 5573 . . . 4 (𝐹𝐴𝐹 ∈ V)
3 eqidd 2739 . . . 4 ((𝐹𝐴𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
42, 3clim 14934 . . 3 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
54ibi 270 . 2 (𝐹𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
65simpld 498 1 (𝐹𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2113  wral 3053  wrex 3054  Vcvv 3397   class class class wbr 5027  cfv 6333  (class class class)co 7164  cc 10606   < clt 10746  cmin 10941  cz 12055  cuz 12317  +crp 12465  abscabs 14676  cli 14924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-cnex 10664  ax-resscn 10665
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7167  df-neg 10944  df-z 12056  df-uz 12318  df-clim 14928
This theorem is referenced by:  rlimclim  14986  climrlim2  14987  climuni  14992  fclim  14993  climeu  14995  climreu  14996  2clim  15012  climcn1lem  15043  climadd  15072  climmul  15073  climsub  15074  climaddc2  15076  climcau  15113  clim2div  15330  ntrivcvgtail  15341  ntrivcvgmullem  15342  mbflim  24413  ulmcau  25134  emcllem6  25730  dchrmusum2  26222  dchrvmasumiflem1  26229  dchrvmasumiflem2  26230  dchrisum0lem1b  26243  dchrmusumlem  26250  iprodefisum  33270  climrec  42670  climexp  42672  climsuse  42675  climneg  42677  climdivf  42679  climleltrp  42743  climuzlem  42810  climxlim2lem  42912  climxlim2  42913  sge0isum  43491
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