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Theorem rlimcl 14176
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 14174 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
2 rlimss 14175 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3 eqidd 2622 . . . 4 ((𝐹𝑟 𝐴𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹𝑥))
41, 2, 3rlim 14168 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦))))
54ibi 256 . 2 (𝐹𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦)))
65simpld 475 1 (𝐹𝑟 𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  wrex 2908   class class class wbr 4618  dom cdm 5079  cfv 5852  (class class class)co 6610  cc 9886  cr 9887   < clt 10026  cle 10027  cmin 10218  +crp 11784  abscabs 13916  𝑟 crli 14158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-pm 7812  df-rlim 14162
This theorem is referenced by:  rlimi  14186  rlimclim1  14218  rlimuni  14223  rlimresb  14238  rlimcld2  14251  rlimabs  14281  rlimcj  14282  rlimre  14283  rlimim  14284  rlimo1  14289  rlimadd  14315  rlimsub  14316  rlimmul  14317  rlimdiv  14318  rlimsqzlem  14321  fsumrlim  14481  dchrisum0lem2a  25123  mulog2sumlem2  25141  mulog2sumlem3  25142
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