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Theorem rlimcl 15539
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 15537 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
2 rlimss 15538 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3 eqidd 2738 . . . 4 ((𝐹𝑟 𝐴𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹𝑥))
41, 2, 3rlim 15531 . . 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 2108  wral 3061  wrex 3070   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431  cc 11153  cr 11154   < clt 11295  cle 11296  cmin 11492  +crp 13034  abscabs 15273  𝑟 crli 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8869  df-rlim 15525
This theorem is referenced by:  rlimi  15549  rlimclim1  15581  rlimuni  15586  rlimresb  15601  rlimcld2  15614  rlimabs  15645  rlimcj  15646  rlimre  15647  rlimim  15648  rlimo1  15653  rlimadd  15679  rlimsub  15680  rlimmul  15681  rlimdiv  15682  rlimsqzlem  15685  fsumrlim  15847  dchrisum0lem2a  27561  mulog2sumlem2  27579  mulog2sumlem3  27580
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