Theorem climcl 11289

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 j k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climrel 11287	. . . . 5 |- Rel ~~>
2	1	brrelex1i 4669	. . . 4 |- ( F ~~> A -> F e. _V )
3		eqidd 2178	. . . 4 |- ( ( F ~~> A /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
4	2, 3	clim 11288	. . 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 176	. 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 112	1 |- ( F ~~> A -> A e. CC )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   < clt 7991   - cmin 8127   ZZcz 9252   ZZ>=cuz 9527   RR+crp 9652   abscabs 11005   ~~> cli 11285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-cnex 7901  ax-resscn 7902

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-neg 8130  df-z 9253  df-uz 9528  df-clim 11286

This theorem is referenced by:  climuni  11300  fclim  11301  climeu  11303  climreu  11304  2clim  11308  climcn1lem  11326  climrecl  11331  climadd  11333  climmul  11334  climsub  11335  climaddc2  11337  climcau  11354  geoisum1c  11527  clim2divap  11547  ntrivcvgap  11555