MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climcl Structured version   Unicode version

Theorem climcl 12295
Description: Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
climcl  |-  ( F  ~~>  A  ->  A  e.  CC )

Proof of Theorem climcl
Dummy variables  x  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climrel 12288 . . . . 5  |-  Rel  ~~>
21brrelexi 4920 . . . 4  |-  ( F  ~~>  A  ->  F  e.  _V )
3 eqidd 2439 . . . 4  |-  ( ( F  ~~>  A  /\  k  e.  ZZ )  ->  ( F `  k )  =  ( F `  k ) )
42, 3clim 12290 . . 3  |-  ( F  ~~>  A  ->  ( F  ~~>  A 
<->  ( A  e.  CC  /\ 
A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
 k )  -  A ) )  < 
x ) ) ) )
54ibi 234 . 2  |-  ( F  ~~>  A  ->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x ) ) )
65simpld 447 1  |-  ( F  ~~>  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   _Vcvv 2958   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990    < clt 9122    - cmin 9293   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   abscabs 12041    ~~> cli 12280
This theorem is referenced by:  rlimclim  12342  climrlim2  12343  climuni  12348  fclim  12349  climeu  12351  climreu  12352  2clim  12368  climcn1lem  12398  climadd  12427  climmul  12428  climsub  12429  climaddc2  12431  climcau  12466  mbflim  19562  ulmcau  20313  emcllem6  20841  dchrmusum2  21190  dchrvmasumiflem1  21197  dchrvmasumiflem2  21198  dchrisum0lem1b  21211  dchrmusumlem  21218  clim2div  25219  ntrivcvgtail  25230  ntrivcvgmullem  25231  iprodefisum  25320  climrec  27707  climexp  27709  climsuse  27712  climneg  27714  climdivf  27716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-cnex 9048  ax-resscn 9049
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-neg 9296  df-z 10285  df-uz 10491  df-clim 12284
  Copyright terms: Public domain W3C validator