Metamath Proof Explorer

Theorem cxpef

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014)

Ref Expression
Assertion cxpef 
|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )

Proof

Step Hyp Ref Expression
1 cxpval 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A ^c B ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )
2 1 3adant2 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )
3 simp2 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> A =/= 0 )
4 3 neneqd 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> -. A = 0 )
5 4 iffalsed 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( B x. ( log ` A ) ) ) )
6 2 5 eqtrd 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )