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	$$⊢\phi \to A\in ℂ$$
	sqrecd.1	$$⊢\phi \to A\ne 0$$
	expclzd.3	$$⊢\phi \to N\in \mathbb{Z}$$
Assertion	exprecd	$$⊢\phi \to {\left(\frac{1}{A}\right)}^N=\frac{1}{A^N}$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		sqrecd.1	$$⊢\phi \to A\ne 0$$
3		expclzd.3	$$⊢\phi \to N\in \mathbb{Z}$$
4		exprec	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge N\in \mathbb{Z}\right)\to {\left(\frac{1}{A}\right)}^N=\frac{1}{A^N}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to {\left(\frac{1}{A}\right)}^N=\frac{1}{A^N}$$