Definition df-cxp 20408

Description: Define the power function on complex numbers. Note that the value of this function when x = 0 and ( Re ` y ) <_ 0 , y =/= 0 should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)

Assertion

Ref	Expression
df-cxp	|- ^ c = ( x e. CC , y e. CC |-> if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-cxp

Step	Hyp	Ref	Expression
1		ccxp 20406	. 2 class ^ c
2		vx	. . 3 set x
3		vy	. . 3 set y
4		cc 8944	. . 3 class CC
5	2	cv 1648	. . . . 5 class x
6		cc0 8946	. . . . 5 class 0
7	5, 6	wceq 1649	. . . 4 wff x = 0
8	3	cv 1648	. . . . . 6 class y
9	8, 6	wceq 1649	. . . . 5 wff y = 0
10		c1 8947	. . . . 5 class 1
11	9, 10, 6	cif 3699	. . . 4 class if ( y = 0 , 1 , 0 )
12		clog 20405	. . . . . . 7 class log
13	5, 12	cfv 5413	. . . . . 6 class ( log ` x )
14		cmul 8951	. . . . . 6 class x.
15	8, 13, 14	co 6040	. . . . 5 class ( y x. ( log ` x ) )
16		ce 12619	. . . . 5 class exp
17	15, 16	cfv 5413	. . . 4 class ( exp ` ( y x. ( log ` x ) ) )
18	7, 11, 17	cif 3699	. . 3 class if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) )
19	2, 3, 4, 4, 18	cmpt2 6042	. 2 class ( x e. CC , y e. CC |-> if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) ) )
20	1, 19	wceq 1649	1 wff ^ c = ( x e. CC , y e. CC |-> if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) ) )

This definition is referenced by:  cxpval  20508