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

Theorem rlimpm 14572
Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimpm (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))

Proof of Theorem rlimpm
Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rlim 14561 . . . . 5 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
2 opabssxp 5398 . . . . 5 {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))} ⊆ ((ℂ ↑pm ℝ) × ℂ)
31, 2eqsstri 3831 . . . 4 𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4 dmss 5526 . . . 4 ( ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ) → dom ⇝𝑟 ⊆ dom ((ℂ ↑pm ℝ) × ℂ))
53, 4ax-mp 5 . . 3 dom ⇝𝑟 ⊆ dom ((ℂ ↑pm ℝ) × ℂ)
6 dmxpss 5782 . . 3 dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
75, 6sstri 3807 . 2 dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8 rlimrel 14565 . . 3 Rel ⇝𝑟
98releldmi 5566 . 2 (𝐹𝑟 𝐴𝐹 ∈ dom ⇝𝑟 )
107, 9sseldi 3796 1 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3089  wrex 3090  wss 3769   class class class wbr 4843  {copab 4905   × cxp 5310  dom cdm 5312  cfv 6101  (class class class)co 6878  pm cpm 8096  cc 10222  cr 10223   < clt 10363  cle 10364  cmin 10556  +crp 12074  abscabs 14315  𝑟 crli 14557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rlim 14561
This theorem is referenced by:  rlimf  14573  rlimss  14574  rlimclim1  14617
  Copyright terms: Public domain W3C validator