Theorem rlimcl 14234

Description: Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
rlimcl	|- ( F ~~> r A -> A e. CC )

Proof of Theorem rlimcl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimf 14232	. . . 4 |- ( F ~~> r A -> F : dom F --> CC )
2		rlimss 14233	. . . 4 |- ( F ~~> r A -> dom F C_ RR )
3		eqidd 2623	. . . 4 |- ( ( F ~~> r A /\ x e. dom F ) -> ( F ` x ) = ( F ` x ) )
4	1, 2, 3	rlim 14226	. . 3 |- ( F ~~> r A -> ( F ~~> r A <-> ( A e. CC /\ A. y e. RR+ E. z e. RR A. x e. dom F ( z <_ x -> ( abs ` ( ( F ` x ) - A ) ) < y ) ) ) )
5	4	ibi 256	. 2 |- ( F ~~> r A -> ( A e. CC /\ A. y e. RR+ E. z e. RR A. x e. dom F ( z <_ x -> ( abs ` ( ( F ` x ) - A ) ) < y ) ) )
6	5	simpld 475	1 |- ( F ~~> r A -> A e. CC )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   abscabs 13974   ~~> r crli 14216

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-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-rlim 14220

This theorem is referenced by:  rlimi  14244  rlimclim1  14276  rlimuni  14281  rlimresb  14296  rlimcld2  14309  rlimabs  14339  rlimcj  14340  rlimre  14341  rlimim  14342  rlimo1  14347  rlimadd  14373  rlimsub  14374  rlimmul  14375  rlimdiv  14376  rlimsqzlem  14379  fsumrlim  14543  dchrisum0lem2a  25206  mulog2sumlem2  25224  mulog2sumlem3  25225