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Theorem rlimpm 15532
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 15521 . . . . 5 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
2 opabssxp 5780 . . . . 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 4029 . . . 4 𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4 dmss 5915 . . . 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 6192 . . 3 dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
75, 6sstri 4004 . 2 dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8 rlimrel 15525 . . 3 Rel ⇝𝑟
98releldmi 5961 . 2 (𝐹𝑟 𝐴𝐹 ∈ dom ⇝𝑟 )
107, 9sselid 3992 1 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wral 3058  wrex 3067  wss 3962   class class class wbr 5147  {copab 5209   × cxp 5686  dom cdm 5688  cfv 6562  (class class class)co 7430  pm cpm 8865  cc 11150  cr 11151   < clt 11292  cle 11293  cmin 11489  +crp 13031  abscabs 15269  𝑟 crli 15517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rlim 15521
This theorem is referenced by:  rlimf  15533  rlimss  15534  rlimclim1  15577
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