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Theorem rlimcl 12260
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 12258 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
2 rlimss 12259 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
3 eqidd 2413 . . . 4  |-  ( ( F  ~~> r  A  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
41, 2, 3rlim 12252 . . 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 233 . 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 446 1  |-  ( F  ~~> r  A  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   A.wral 2674   E.wrex 2675   class class class wbr 4180   dom cdm 4845   ` cfv 5421  (class class class)co 6048   CCcc 8952   RRcr 8953    < clt 9084    <_ cle 9085    - cmin 9255   RR+crp 10576   abscabs 12002    ~~> r crli 12242
This theorem is referenced by:  rlimi  12270  rlimclim1  12302  rlimuni  12307  rlimresb  12322  rlimcld2  12335  rlimabs  12365  rlimcj  12366  rlimre  12367  rlimim  12368  rlimo1  12373  rlimadd  12399  rlimsub  12400  rlimmul  12401  rlimdiv  12402  rlimsqzlem  12405  fsumrlim  12553  dchrisum0lem2a  21172  mulog2sumlem2  21190  mulog2sumlem3  21191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-pm 6988  df-rlim 12246
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