Theorem absexpd

Description: Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	|- ( ph -> A e. CC )
	absexpd.2	|- ( ph -> N e. NN0 )
Assertion	absexpd	|- ( ph -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2		absexpd.2	|- ( ph -> N e. NN0 )
3		absexp	|- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
4	1 2 3	syl2anc	|- ( ph -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )