Theorem climcl 11245

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 11243	. . . . 5 |- Rel ~~>
2	1	brrelex1i 4654	. . . 4 |- ( F ~~> A -> F e. _V )
3		eqidd 2171	. . . 4 |- ( ( F ~~> A /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
4	2, 3	clim 11244	. . 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 175	. 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 111	1 |- ( F ~~> A -> A e. CC )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   E.wrex 2449   _Vcvv 2730   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   < clt 7954   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   abscabs 10961   ~~> cli 11241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-neg 8093  df-z 9213  df-uz 9488  df-clim 11242

This theorem is referenced by:  climuni  11256  fclim  11257  climeu  11259  climreu  11260  2clim  11264  climcn1lem  11282  climrecl  11287  climadd  11289  climmul  11290  climsub  11291  climaddc2  11293  climcau  11310  geoisum1c  11483  clim2divap  11503  ntrivcvgap  11511