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Theorem rlimcl 13326
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

Proof of Theorem rlimcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimf 13324 . . . 4
2 rlimss 13325 . . . 4
3 eqidd 2458 . . . 4
41, 2, 3rlim 13318 . . 3
54ibi 241 . 2
65simpld 459 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  E.wrex 2808   class class class wbr 4452  domcdm 5004  `cfv 5593  (class class class)co 6296   cc 9511   cr 9512   clt 9649   cle 9650   cmin 9828   crp 11249   cabs 13067   crli 13308
This theorem is referenced by:  rlimi  13336  rlimclim1  13368  rlimuni  13373  rlimresb  13388  rlimcld2  13401  rlimabs  13431  rlimcj  13432  rlimre  13433  rlimim  13434  rlimo1  13439  rlimadd  13465  rlimsub  13466  rlimmul  13467  rlimdiv  13468  rlimsqzlem  13471  fsumrlim  13625  dchrisum0lem2a  23702  mulog2sumlem2  23720  mulog2sumlem3  23721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-cnex 9569  ax-resscn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-pm 7442  df-rlim 13312
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