Theorem expnegd

Description: Value of a nonzero complex number raised to the negative of an integer power. (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	expnegd	$$⊢\phi \to A^{-N}=\frac{1}{A^N}$$

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		expnegz	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge N\in \mathbb{Z}\right)\to A^{-N}=\frac{1}{A^N}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^{-N}=\frac{1}{A^N}$$