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Theorem climcl 11019
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 11017 . . . . 5  |-  Rel  ~~>
21brrelex1i 4552 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2118 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 11018 . . 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 175 . 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 111 1  |-  ( F  ~~>  A  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   CCcc 7586    < clt 7768    - cmin 7901   ZZcz 9022   ZZ>=cuz 9294   RR+crp 9409   abscabs 10737    ~~> cli 11015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-cnex 7679  ax-resscn 7680
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-neg 7904  df-z 9023  df-uz 9295  df-clim 11016
This theorem is referenced by:  climuni  11030  fclim  11031  climeu  11033  climreu  11034  2clim  11038  climcn1lem  11056  climrecl  11061  climadd  11063  climmul  11064  climsub  11065  climaddc2  11067  climcau  11084  geoisum1c  11257
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