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Theorem climcl 13471
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  |-  ( F  ~~>  A  ->  A  e.  CC )

Proof of Theorem climcl
Dummy variables  x  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climrel 13464 . . . . 5  |-  Rel  ~~>
21brrelexi 4864 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2403 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 13466 . . 3  |-  ( F  ~~>  A  ->  ( F  ~~>  A 
<->  ( A  e.  CC  /\ 
A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
 k )  -  A ) )  < 
x ) ) ) )
54ibi 241 . 2  |-  ( F  ~~>  A  ->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x ) ) )
65simpld 457 1  |-  ( F  ~~>  A  ->  A  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   CCcc 9520    < clt 9658    - cmin 9841   ZZcz 10905   ZZ>=cuz 11127   RR+crp 11265   abscabs 13216    ~~> cli 13456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-cnex 9578  ax-resscn 9579
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-neg 9844  df-z 10906  df-uz 11128  df-clim 13460
This theorem is referenced by:  rlimclim  13518  climrlim2  13519  climuni  13524  fclim  13525  climeu  13527  climreu  13528  2clim  13544  climcn1lem  13574  climadd  13603  climmul  13604  climsub  13605  climaddc2  13607  climcau  13642  clim2div  13850  ntrivcvgtail  13861  ntrivcvgmullem  13862  mbflim  22367  ulmcau  23082  emcllem6  23656  dchrmusum2  24060  dchrvmasumiflem1  24067  dchrvmasumiflem2  24068  dchrisum0lem1b  24081  dchrmusumlem  24088  iprodefisum  29950  climrec  36977  climexp  36979  climsuse  36982  climneg  36984  climdivf  36986
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