Definition df-ef 15758

Description: Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 15770. (Contributed by NM, 14-Mar-2005.)

Assertion

Ref	Expression
df-ef	⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))

Distinct variable group:   𝑥,𝑘

Detailed syntax breakdown of Definition df-ef

Step	Hyp	Ref	Expression
1		ce 15752	. 2 class exp
2		vx	. . 3 setvar 𝑥
3		cc 10853	. . 3 class ℂ
4		cn0 12216	. . . 4 class ℕ0
5	2	cv 1540	. . . . . 6 class 𝑥
6		vk	. . . . . . 7 setvar 𝑘
7	6	cv 1540	. . . . . 6 class 𝑘
8		cexp 13763	. . . . . 6 class ↑
9	5, 7, 8	co 7268	. . . . 5 class (𝑥↑𝑘)
10		cfa 13968	. . . . . 6 class !
11	7, 10	cfv 6430	. . . . 5 class (!‘𝑘)
12		cdiv 11615	. . . . 5 class /
13	9, 11, 12	co 7268	. . . 4 class ((𝑥↑𝑘) / (!‘𝑘))
14	4, 13, 6	csu 15378	. . 3 class Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘))
15	2, 3, 14	cmpt 5161	. 2 class (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
16	1, 15	wceq 1541	1 wff exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))

This definition is referenced by:  efval  15770  eff  15772