Theorem climcl 14230

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.

Step	Hyp	Ref	Expression
1		climrel 14223	. . . . 5 |- Rel ~~>
2	1	brrelexi 5158	. . . 4 |- ( F ~~> A -> F e. _V )
3		eqidd 2623	. . . 4 |- ( ( F ~~> A /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
4	2, 3	clim 14225	. . 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 ) ) ) )
5	4	ibi 256	. 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 ) ) )
6	5	simpld 475	1 |- ( F ~~> A -> A e. CC )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  rlimclim  14277  climrlim2  14278  climuni  14283  fclim  14284  climeu  14286  climreu  14287  2clim  14303  climcn1lem  14333  climadd  14362  climmul  14363  climsub  14364  climaddc2  14366  climcau  14401  clim2div  14621  ntrivcvgtail  14632  ntrivcvgmullem  14633  mbflim  23435  ulmcau  24149  emcllem6  24727  dchrmusum2  25183  dchrvmasumiflem1  25190  dchrvmasumiflem2  25191  dchrisum0lem1b  25204  dchrmusumlem  25211  iprodefisum  31627  climrec  39835  climexp  39837  climsuse  39840  climneg  39842  climdivf  39844  climleltrp  39908  climuzlem  39975  climxlim2lem  40071  climxlim2  40072  sge0isum  40644