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Theorem climcl 13281
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 13274 . . . . 5  |-  Rel  ~~>
21brrelexi 5039 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2468 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 13276 . . 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 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486    < clt 9624    - cmin 9801   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   abscabs 13026    ~~> cli 13266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-cnex 9544  ax-resscn 9545
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-neg 9804  df-z 10861  df-uz 11079  df-clim 13270
This theorem is referenced by:  rlimclim  13328  climrlim2  13329  climuni  13334  fclim  13335  climeu  13337  climreu  13338  2clim  13354  climcn1lem  13384  climadd  13413  climmul  13414  climsub  13415  climaddc2  13417  climcau  13452  mbflim  21810  ulmcau  22524  emcllem6  23058  dchrmusum2  23407  dchrvmasumiflem1  23414  dchrvmasumiflem2  23415  dchrisum0lem1b  23428  dchrmusumlem  23435  clim2div  28600  ntrivcvgtail  28611  ntrivcvgmullem  28612  iprodefisum  28701  climrec  31145  climexp  31147  climsuse  31150  climneg  31152  climdivf  31154
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