MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climcl Structured version   Visualization version   GIF version

Theorem climcl 14855
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 14848 . . . . 5 Rel ⇝
21brrelex1i 5607 . . . 4 (𝐹𝐴𝐹 ∈ V)
3 eqidd 2822 . . . 4 ((𝐹𝐴𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
42, 3clim 14850 . . 3 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
54ibi 269 . 2 (𝐹𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
65simpld 497 1 (𝐹𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110  wral 3138  wrex 3139  Vcvv 3494   class class class wbr 5065  cfv 6354  (class class class)co 7155  cc 10534   < clt 10674  cmin 10869  cz 11980  cuz 12242  +crp 12388  abscabs 14592  cli 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-cnex 10592  ax-resscn 10593
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-neg 10872  df-z 11981  df-uz 12243  df-clim 14844
This theorem is referenced by:  rlimclim  14902  climrlim2  14903  climuni  14908  fclim  14909  climeu  14911  climreu  14912  2clim  14928  climcn1lem  14958  climadd  14987  climmul  14988  climsub  14989  climaddc2  14991  climcau  15026  clim2div  15244  ntrivcvgtail  15255  ntrivcvgmullem  15256  mbflim  24268  ulmcau  24982  emcllem6  25577  dchrmusum2  26069  dchrvmasumiflem1  26076  dchrvmasumiflem2  26077  dchrisum0lem1b  26090  dchrmusumlem  26097  iprodefisum  32973  climrec  41884  climexp  41886  climsuse  41889  climneg  41891  climdivf  41893  climleltrp  41957  climuzlem  42024  climxlim2lem  42126  climxlim2  42127  sge0isum  42710
  Copyright terms: Public domain W3C validator