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Theorem rlimcl 12302
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 12300 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
2 rlimss 12301 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
3 eqidd 2439 . . . 4  |-  ( ( F  ~~> r  A  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
41, 2, 3rlim 12294 . . 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 234 . 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 447 1  |-  ( F  ~~> r  A  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4215   dom cdm 4881   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994    < clt 9125    <_ cle 9126    - cmin 9296   RR+crp 10617   abscabs 12044    ~~> r crli 12284
This theorem is referenced by:  rlimi  12312  rlimclim1  12344  rlimuni  12349  rlimresb  12364  rlimcld2  12377  rlimabs  12407  rlimcj  12408  rlimre  12409  rlimim  12410  rlimo1  12415  rlimadd  12441  rlimsub  12442  rlimmul  12443  rlimdiv  12444  rlimsqzlem  12447  fsumrlim  12595  dchrisum0lem2a  21216  mulog2sumlem2  21234  mulog2sumlem3  21235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-pm 7024  df-rlim 12288
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