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Theorem climcl 12961
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 12954 . . . . 5  |-  Rel  ~~>
21brrelexi 4866 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2434 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 12956 . . 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 456 1  |-  ( F  ~~>  A  ->  A  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9268    < clt 9406    - cmin 9583   ZZcz 10634   ZZ>=cuz 10849   RR+crp 10979   abscabs 12707    ~~> cli 12946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-cnex 9326  ax-resscn 9327
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-ov 6083  df-neg 9586  df-z 10635  df-uz 10850  df-clim 12950
This theorem is referenced by:  rlimclim  13008  climrlim2  13009  climuni  13014  fclim  13015  climeu  13017  climreu  13018  2clim  13034  climcn1lem  13064  climadd  13093  climmul  13094  climsub  13095  climaddc2  13097  climcau  13132  mbflim  20988  ulmcau  21745  emcllem6  22279  dchrmusum2  22628  dchrvmasumiflem1  22635  dchrvmasumiflem2  22636  dchrisum0lem1b  22649  dchrmusumlem  22656  clim2div  27251  ntrivcvgtail  27262  ntrivcvgmullem  27263  iprodefisum  27352  climrec  29622  climexp  29624  climsuse  29627  climneg  29629  climdivf  29631
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