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Theorem climcl 13081
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 13074 . . . . 5  |-  Rel  ~~>
21brrelexi 4979 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2452 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 13076 . . 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 459 1  |-  ( F  ~~>  A  ->  A  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   CCcc 9383    < clt 9521    - cmin 9698   ZZcz 10749   ZZ>=cuz 10964   RR+crp 11094   abscabs 12827    ~~> cli 13066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-cnex 9441  ax-resscn 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-neg 9701  df-z 10750  df-uz 10965  df-clim 13070
This theorem is referenced by:  rlimclim  13128  climrlim2  13129  climuni  13134  fclim  13135  climeu  13137  climreu  13138  2clim  13154  climcn1lem  13184  climadd  13213  climmul  13214  climsub  13215  climaddc2  13217  climcau  13252  mbflim  21264  ulmcau  21978  emcllem6  22512  dchrmusum2  22861  dchrvmasumiflem1  22868  dchrvmasumiflem2  22869  dchrisum0lem1b  22882  dchrmusumlem  22889  clim2div  27540  ntrivcvgtail  27551  ntrivcvgmullem  27552  iprodefisum  27641  climrec  29916  climexp  29918  climsuse  29921  climneg  29923  climdivf  29925
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