Definition df-ef 12415

Description: Define the exponential function. Its value at the complex number A is ( exp ` A ) and is called the "exponential of A"; see efval 12428. (Contributed by NM, 14-Mar-2005.)

Assertion

Ref	Expression
df-ef	|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )

Distinct variable group:   x, k

Detailed syntax breakdown of Definition df-ef

Step	Hyp	Ref	Expression
1		ce 12409	. 2 class exp
2		vx	. . 3 setvar x
3		cc 8177	. . 3 class CC
4		cn0 9563	. . . 4 class NN0
5	2	cv 1401	. . . . . 6 class x
6		vk	. . . . . . 7 setvar k
7	6	cv 1401	. . . . . 6 class k
8		cexp 10975	. . . . . 6 class ^
9	5, 7, 8	co 6085	. . . . 5 class ( x ^ k )
10		cfa 11163	. . . . . 6 class !
11	7, 10	cfv 5377	. . . . 5 class ( ! ` k )
12		cdiv 9002	. . . . 5 class /
13	9, 11, 12	co 6085	. . . 4 class ( ( x ^ k ) / ( ! ` k ) )
14	4, 13, 6	csu 12119	. . 3 class sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) )
15	2, 3, 14	cmpt 4192	. 2 class ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
16	1, 15	wceq 1402	1 wff exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )

This definition is used by:  efval  12428  eff  12430