Theorem climcl 12048

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 12046	. . . . 5 |- Rel ~~>
2	1	brrelex1i 4818	. . . 4 |- ( F ~~> A -> F e. _V )
3		eqidd 2239	. . . 4 |- ( ( F ~~> A /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
4	2, 3	clim 12047	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   < clt 8360   - cmin 8497   ZZcz 9644   ZZ>=cuz 9921   RR+crp 10054   abscabs 11763   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-neg 8500  df-z 9645  df-uz 9922  df-clim 12045

This theorem is used by:  climuni  12059  fclim  12060  climeu  12062  climreu  12063  2clim  12067  climcn1lem  12085  climrecl  12090  climadd  12092  climmul  12093  climsub  12094  climaddc2  12096  climcau  12113  geoisum1c  12287  clim2divap  12307  ntrivcvgap  12315