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Theorem rlimcl 13338
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 13336 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
2 rlimss 13337 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
3 eqidd 2458 . . . 4  |-  ( ( F  ~~> r  A  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
41, 2, 3rlim 13330 . . 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 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   dom cdm 5008   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508    < clt 9645    <_ cle 9646    - cmin 9824   RR+crp 11245   abscabs 13079    ~~> r crli 13320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-pm 7441  df-rlim 13324
This theorem is referenced by:  rlimi  13348  rlimclim1  13380  rlimuni  13385  rlimresb  13400  rlimcld2  13413  rlimabs  13443  rlimcj  13444  rlimre  13445  rlimim  13446  rlimo1  13451  rlimadd  13477  rlimsub  13478  rlimmul  13479  rlimdiv  13480  rlimsqzlem  13483  fsumrlim  13637  dchrisum0lem2a  23828  mulog2sumlem2  23846  mulog2sumlem3  23847
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