Theorem absexpd

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

	Ref	Expression
Hypotheses	abscld.1	$$⊢\phi \to A\in ℂ$$
	absexpd.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	absexpd	$$⊢\phi \to \left|A^N\right|={\left|A\right|}^N$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		absexpd.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		absexp	$$⊢\left(A\in ℂ\wedge N\in {\mathbb{N}}_0\right)\to \left|A^N\right|={\left|A\right|}^N$$
4	1 2 3	syl2anc	$$⊢\phi \to \left|A^N\right|={\left|A\right|}^N$$