Theorem exprecd

Description: An integer power of a reciprocal is the reciprocal of the integer power with same exponent. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	|- ( ph -> A e. CC )
	sqrecd.1	|- ( ph -> A =/= 0 )
	expclzd.3	|- ( ph -> N e. ZZ )
Assertion	exprecd	|- ( ph -> ( ( 1 / A ) ^ N ) = ( 1 / ( A ^ N ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	|- ( ph -> A e. CC )
2		sqrecd.1	|- ( ph -> A =/= 0 )
3		expclzd.3	|- ( ph -> N e. ZZ )
4		exprec	|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( ( 1 / A ) ^ N ) = ( 1 / ( A ^ N ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( 1 / A ) ^ N ) = ( 1 / ( A ^ N ) ) )