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

Theorem rlimcl 14836
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 14834 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
2 rlimss 14835 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3 eqidd 2821 . . . 4 ((𝐹𝑟 𝐴𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹𝑥))
41, 2, 3rlim 14828 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦))))
54ibi 269 . 2 (𝐹𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧𝑥 → (abs‘((𝐹𝑥) − 𝐴)) < 𝑦)))
65simpld 497 1 (𝐹𝑟 𝐴𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2114  wral 3125  wrex 3126   class class class wbr 5038  dom cdm 5527  cfv 6327  (class class class)co 7129  cc 10509  cr 10510   < clt 10649  cle 10650  cmin 10844  +crp 12364  abscabs 14569  𝑟 crli 14818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-cnex 10567  ax-resscn 10568
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fv 6335  df-ov 7132  df-oprab 7133  df-mpo 7134  df-pm 8383  df-rlim 14822
This theorem is referenced by:  rlimi  14846  rlimclim1  14878  rlimuni  14883  rlimresb  14898  rlimcld2  14911  rlimabs  14941  rlimcj  14942  rlimre  14943  rlimim  14944  rlimo1  14949  rlimadd  14975  rlimsub  14976  rlimmul  14977  rlimdiv  14978  rlimsqzlem  14981  fsumrlim  15142  dchrisum0lem2a  26076  mulog2sumlem2  26094  mulog2sumlem3  26095
  Copyright terms: Public domain W3C validator