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Definition df-ef 15822
Description: Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 15834. (Contributed by NM, 14-Mar-2005.)
Assertion
Ref Expression
df-ef exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))
Distinct variable group:   𝑥,𝑘

Detailed syntax breakdown of Definition df-ef
StepHypRef Expression
1 ce 15816 . 2 class exp
2 vx . . 3 setvar 𝑥
3 cc 10915 . . 3 class
4 cn0 12279 . . . 4 class 0
52cv 1538 . . . . . 6 class 𝑥
6 vk . . . . . . 7 setvar 𝑘
76cv 1538 . . . . . 6 class 𝑘
8 cexp 13828 . . . . . 6 class
95, 7, 8co 7307 . . . . 5 class (𝑥𝑘)
10 cfa 14033 . . . . . 6 class !
117, 10cfv 6458 . . . . 5 class (!‘𝑘)
12 cdiv 11678 . . . . 5 class /
139, 11, 12co 7307 . . . 4 class ((𝑥𝑘) / (!‘𝑘))
144, 13, 6csu 15442 . . 3 class Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘))
152, 3, 14cmpt 5164 . 2 class (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))
161, 15wceq 1539 1 wff exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))
Colors of variables: wff setvar class
This definition is referenced by:  efval  15834  eff  15836
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