Theorem rlimpm 15423

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 15412	. . . . 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 5716	. . . . 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 3980	. . . 4 ⊢ ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ)
4		dmss 5851	. . . 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 6129	. . 3 ⊢ dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ)
7	5, 6	sstri 3943	. 2 ⊢ dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ)
8		rlimrel 15416	. . 3 ⊢ Rel ⇝𝑟
9	8	releldmi 5897	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ dom ⇝𝑟 )
10	7, 9	sselid 3931	1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901   class class class wbr 5098  {copab 5160   × cxp 5622  dom cdm 5624  ‘cfv 6492  (class class class)co 7358   ↑_pm cpm 8764  ℂcc 11024  ℝcr 11025   < clt 11166   ≤ cle 11167   − cmin 11364  ℝ⁺crp 12905  abscabs 15157   ⇝_𝑟 crli 15408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-rlim 15412

This theorem is referenced by:  rlimf  15424  rlimss  15425  rlimclim1  15468