Theorem cxprecd

Description: Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	rpcxpcld.1	|- ( ph -> A e. RR+ )
	cxprecd.2	|- ( ph -> B e. CC )
Assertion	cxprecd	|- ( ph -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	|- ( ph -> A e. RR+ )
2		cxprecd.2	|- ( ph -> B e. CC )
3		cxprec	|- ( ( A e. RR+ /\ B e. CC ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )