Theorem rlimpm 15577

Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Assertion

Ref	Expression
rlimpm	⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ))

Proof of Theorem rlimpm

Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rlim 15566	. . . . 5 ⊢ ⇝𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))}
2		opabssxp 5758	. . . . 5 ⊢ {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))} ⊆ ((ℂ ↑pm ℝ) × ℂ)
3	1, 2	eqsstri 3986	. . . 4 ⊢ ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4		dmss 5897	. . . 4 ⊢ ( ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ) → dom ⇝𝑟 ⊆ dom ((ℂ ↑pm ℝ) × ℂ))
5	3, 4	ax-mp 5	. . 3 ⊢ dom ⇝𝑟 ⊆ dom ((ℂ ↑pm ℝ) × ℂ)
6		dmxpss 6174	. . 3 ⊢ dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
7	5, 6	sstri 3949	. 2 ⊢ dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8		rlimrel 15570	. . 3 ⊢ Rel ⇝𝑟
9	8	releldmi 5943	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ dom ⇝𝑟 )
10	7, 9	sselid 3938	1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908   class class class wbr 5114  {copab 5178   × cxp 5664  dom cdm 5666  ‘cfv 6543  (class class class)co 7423   ↑_pm cpm 8834  ℂcc 11116  ℝcr 11117   < clt 11261   ≤ cle 11262   − cmin 11459  ℝ⁺crp 13034  abscabs 15311   ⇝_𝑟 crli 15562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-dm 5676  df-rlim 15566

This theorem is used by:  rlimf  15578  rlimss  15579  rlimclim1  15622