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Theorem rlimpm 15536
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 15525 . . . . 5 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
2 opabssxp 5778 . . . . 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 4030 . . . 4 𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4 dmss 5913 . . . 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 6191 . . 3 dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
75, 6sstri 3993 . 2 dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8 rlimrel 15529 . . 3 Rel ⇝𝑟
98releldmi 5959 . 2 (𝐹𝑟 𝐴𝐹 ∈ dom ⇝𝑟 )
107, 9sselid 3981 1 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  wrex 3070  wss 3951   class class class wbr 5143  {copab 5205   × cxp 5683  dom cdm 5685  cfv 6561  (class class class)co 7431  pm cpm 8867  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-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rlim 15525
This theorem is referenced by:  rlimf  15537  rlimss  15538  rlimclim1  15581
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