Theorem cxpexpz

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpexpz	|- ( ( A e. CC /\ A =/= 0 /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )

Proof

Step	Hyp	Ref	Expression
1		zcn	|- ( B e. ZZ -> B e. CC )
2		cxpef	|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
3	1 2	syl3an3	|- ( ( A e. CC /\ A =/= 0 /\ B e. ZZ ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
4		explog	|- ( ( A e. CC /\ A =/= 0 /\ B e. ZZ ) -> ( A ^ B ) = ( exp ` ( B x. ( log ` A ) ) ) )
5	3 4	eqtr4d	|- ( ( A e. CC /\ A =/= 0 /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )