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Theorem rlimcl 12986
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 12984 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
2 rlimss 12985 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
3 eqidd 2444 . . . 4  |-  ( ( F  ~~> r  A  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
41, 2, 3rlim 12978 . . 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 241 . 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 459 1  |-  ( F  ~~> r  A  ->  A  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2720   E.wrex 2721   class class class wbr 4297   dom cdm 4845   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286    < clt 9423    <_ cle 9424    - cmin 9600   RR+crp 10996   abscabs 12728    ~~> r crli 12968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-pm 7222  df-rlim 12972
This theorem is referenced by:  rlimi  12996  rlimclim1  13028  rlimuni  13033  rlimresb  13048  rlimcld2  13061  rlimabs  13091  rlimcj  13092  rlimre  13093  rlimim  13094  rlimo1  13099  rlimadd  13125  rlimsub  13126  rlimmul  13127  rlimdiv  13128  rlimsqzlem  13131  fsumrlim  13279  dchrisum0lem2a  22771  mulog2sumlem2  22789  mulog2sumlem3  22790
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