Theorem cxpexpzd

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	|- ( ph -> A e. CC )
	cxpefd.2	|- ( ph -> A =/= 0 )
	cxpexpzd.3	|- ( ph -> B e. ZZ )
Assertion	cxpexpzd	|- ( ph -> ( A ^c B ) = ( A ^ B ) )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	|- ( ph -> A e. CC )
2		cxpefd.2	|- ( ph -> A =/= 0 )
3		cxpexpzd.3	|- ( ph -> B e. ZZ )
4		cxpexpz	|- ( ( A e. CC /\ A =/= 0 /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A ^c B ) = ( A ^ B ) )