Theorem rlimpm 15425

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 15414	. . . . 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 5715	. . . . 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 3984	. . . 4 ⊢ ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4		dmss 5849	. . . 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 6124	. . 3 ⊢ dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
7	5, 6	sstri 3947	. 2 ⊢ dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8		rlimrel 15418	. . 3 ⊢ Rel ⇝𝑟
9	8	releldmi 5894	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ dom ⇝𝑟 )
10	7, 9	sselid 3935	1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095  {copab 5157   × cxp 5621  dom cdm 5623  ‘cfv 6486  (class class class)co 7353   ↑_pm cpm 8761  ℂcc 11026  ℝcr 11027   < clt 11168   ≤ cle 11169   − cmin 11365  ℝ⁺crp 12911  abscabs 15159   ⇝_𝑟 crli 15410

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rlim 15414

This theorem is referenced by:  rlimf  15426  rlimss  15427  rlimclim1  15470