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Definition df-cpn 23853
Description: Define the set of 𝑛-times continuously differentiable functions. (Contributed by Stefan O'Rear, 15-Nov-2014.)
Assertion
Ref Expression
df-cpn Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}))
Distinct variable group:   𝑓,𝑠,𝑥

Detailed syntax breakdown of Definition df-cpn
StepHypRef Expression
1 ccpn 23849 . 2 class Cn
2 vs . . 3 setvar 𝑠
3 cc 10136 . . . 4 class
43cpw 4297 . . 3 class 𝒫 ℂ
5 vx . . . 4 setvar 𝑥
6 cn0 11494 . . . 4 class 0
75cv 1630 . . . . . . 7 class 𝑥
82cv 1630 . . . . . . . 8 class 𝑠
9 vf . . . . . . . . 9 setvar 𝑓
109cv 1630 . . . . . . . 8 class 𝑓
11 cdvn 23848 . . . . . . . 8 class D𝑛
128, 10, 11co 6793 . . . . . . 7 class (𝑠 D𝑛 𝑓)
137, 12cfv 6031 . . . . . 6 class ((𝑠 D𝑛 𝑓)‘𝑥)
1410cdm 5249 . . . . . . 7 class dom 𝑓
15 ccncf 22899 . . . . . . 7 class cn
1614, 3, 15co 6793 . . . . . 6 class (dom 𝑓cn→ℂ)
1713, 16wcel 2145 . . . . 5 wff ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)
18 cpm 8010 . . . . . 6 class pm
193, 8, 18co 6793 . . . . 5 class (ℂ ↑pm 𝑠)
2017, 9, 19crab 3065 . . . 4 class {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}
215, 6, 20cmpt 4863 . . 3 class (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)})
222, 4, 21cmpt 4863 . 2 class (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}))
231, 22wceq 1631 1 wff Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}))
Colors of variables: wff setvar class
This definition is referenced by:  cpnfval  23915
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