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Theorem rlimpm 14608
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 14597 . . . . 5 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
2 opabssxp 5428 . . . . 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 3860 . . . 4 𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4 dmss 5555 . . . 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 5806 . . 3 dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
75, 6sstri 3836 . 2 dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8 rlimrel 14601 . . 3 Rel ⇝𝑟
98releldmi 5595 . 2 (𝐹𝑟 𝐴𝐹 ∈ dom ⇝𝑟 )
107, 9sseldi 3825 1 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2166  wral 3117  wrex 3118  wss 3798   class class class wbr 4873  {copab 4935   × cxp 5340  dom cdm 5342  cfv 6123  (class class class)co 6905  pm cpm 8123  cc 10250  cr 10251   < clt 10391  cle 10392  cmin 10585  +crp 12112  abscabs 14351  𝑟 crli 14593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-dm 5352  df-rlim 14597
This theorem is referenced by:  rlimf  14609  rlimss  14610  rlimclim1  14653
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