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Theorem rlimcl 15096
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 15094 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
2 rlimss 15095 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3 eqidd 2740 . . . 4 ((𝐹𝑟 𝐴𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹𝑥))
41, 2, 3rlim 15088 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦))))
54ibi 270 . 2 (𝐹𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦)))
65simpld 498 1 (𝐹𝑟 𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  wral 3064  wrex 3065   class class class wbr 5069  dom cdm 5568  cfv 6400  (class class class)co 7234  cc 10756  cr 10757   < clt 10896  cle 10897  cmin 11091  +crp 12615  abscabs 14829  𝑟 crli 15078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544  ax-cnex 10814  ax-resscn 10815
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-pm 8534  df-rlim 15082
This theorem is referenced by:  rlimi  15106  rlimclim1  15138  rlimuni  15143  rlimresb  15158  rlimcld2  15171  rlimabs  15202  rlimcj  15203  rlimre  15204  rlimim  15205  rlimo1  15210  rlimadd  15236  rlimaddOLD  15237  rlimsub  15238  rlimmul  15239  rlimmulOLD  15240  rlimdiv  15241  rlimsqzlem  15244  fsumrlim  15407  dchrisum0lem2a  26429  mulog2sumlem2  26447  mulog2sumlem3  26448
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