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Theorem climcl 11290
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  j  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climrel 11288 . . . . 5  |-  Rel  ~~>
21brrelex1i 4670 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2178 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 11289 . . 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 176 . 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 112 1  |-  ( F  ~~>  A  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2738   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   CCcc 7809    < clt 7992    - cmin 8128   ZZcz 9253   ZZ>=cuz 9528   RR+crp 9653   abscabs 11006    ~~> cli 11286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-cnex 7902  ax-resscn 7903
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-neg 8131  df-z 9254  df-uz 9529  df-clim 11287
This theorem is referenced by:  climuni  11301  fclim  11302  climeu  11304  climreu  11305  2clim  11309  climcn1lem  11327  climrecl  11332  climadd  11334  climmul  11335  climsub  11336  climaddc2  11338  climcau  11355  geoisum1c  11528  clim2divap  11548  ntrivcvgap  11556
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