MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimcl Structured version   Visualization version   GIF version

Theorem rlimcl 15535
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 (𝐹𝑟 𝐴𝐴 ∈ ℂ)

Proof of Theorem rlimcl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimf 15533 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
2 rlimss 15534 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3 eqidd 2735 . . . 4 ((𝐹𝑟 𝐴𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹𝑥))
41, 2, 3rlim 15527 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦))))
54ibi 267 . 2 (𝐹𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦)))
65simpld 494 1 (𝐹𝑟 𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wral 3058  wrex 3067   class class class wbr 5147  dom cdm 5688  cfv 6562  (class class class)co 7430  cc 11150  cr 11151   < clt 11292  cle 11293  cmin 11489  +crp 13031  abscabs 15269  𝑟 crli 15517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-pm 8867  df-rlim 15521
This theorem is referenced by:  rlimi  15545  rlimclim1  15577  rlimuni  15582  rlimresb  15597  rlimcld2  15610  rlimabs  15641  rlimcj  15642  rlimre  15643  rlimim  15644  rlimo1  15649  rlimadd  15675  rlimsub  15676  rlimmul  15677  rlimdiv  15678  rlimsqzlem  15681  fsumrlim  15843  dchrisum0lem2a  27575  mulog2sumlem2  27593  mulog2sumlem3  27594
  Copyright terms: Public domain W3C validator