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Theorem rlimpm 14836
 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 14825 . . . . 5 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
2 opabssxp 5616 . . . . 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 3977 . . . 4 𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4 dmss 5744 . . . 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 6001 . . 3 dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
75, 6sstri 3952 . 2 dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8 rlimrel 14829 . . 3 Rel ⇝𝑟
98releldmi 5791 . 2 (𝐹𝑟 𝐴𝐹 ∈ dom ⇝𝑟 )
107, 9sseldi 3941 1 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115  ∀wral 3126  ∃wrex 3127   ⊆ wss 3910   class class class wbr 5039  {copab 5101   × cxp 5526  dom cdm 5528  ‘cfv 6328  (class class class)co 7130   ↑pm cpm 8382  ℂcc 10512  ℝcr 10513   < clt 10652   ≤ cle 10653   − cmin 10847  ℝ+crp 12367  abscabs 14572   ⇝𝑟 crli 14821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rlim 14825 This theorem is referenced by:  rlimf  14837  rlimss  14838  rlimclim1  14881
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