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Theorem rlimcl 12073
Description: Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
rlimcl  |-  ( F  ~~> r  A  ->  A  e.  CC )

Proof of Theorem rlimcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimf 12071 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
2 rlimss 12072 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
3 eqidd 2359 . . . 4  |-  ( ( F  ~~> r  A  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
41, 2, 3rlim 12065 . . 3  |-  ( F  ~~> r  A  ->  ( F 
~~> r  A  <->  ( A  e.  CC  /\  A. y  e.  RR+  E. z  e.  RR  A. x  e. 
dom  F ( z  <_  x  ->  ( abs `  ( ( F `
 x )  -  A ) )  < 
y ) ) ) )
54ibi 232 . 2  |-  ( F  ~~> r  A  ->  ( A  e.  CC  /\  A. y  e.  RR+  E. z  e.  RR  A. x  e. 
dom  F ( z  <_  x  ->  ( abs `  ( ( F `
 x )  -  A ) )  < 
y ) ) )
65simpld 445 1  |-  ( F  ~~> r  A  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   A.wral 2619   E.wrex 2620   class class class wbr 4104   dom cdm 4771   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826    < clt 8957    <_ cle 8958    - cmin 9127   RR+crp 10446   abscabs 11815    ~~> r crli 12055
This theorem is referenced by:  rlimi  12083  rlimclim1  12115  rlimuni  12120  rlimresb  12135  rlimcld2  12148  rlimabs  12178  rlimcj  12179  rlimre  12180  rlimim  12181  rlimo1  12186  rlimadd  12212  rlimsub  12213  rlimmul  12214  rlimdiv  12215  rlimsqzlem  12218  fsumrlim  12366  dchrisum0lem2a  20778  mulog2sumlem2  20796  mulog2sumlem3  20797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-pm 6863  df-rlim 12059
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