Theorem cxpnegd

Description: Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	$$⊢\phi \to A\in ℂ$$
	cxpefd.2	$$⊢\phi \to A\ne 0$$
	cxpefd.3	$$⊢\phi \to B\in ℂ$$
Assertion	cxpnegd	$$⊢\phi \to A^{-B}=\frac{1}{A^B}$$

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	$$⊢\phi \to A\in ℂ$$
2		cxpefd.2	$$⊢\phi \to A\ne 0$$
3		cxpefd.3	$$⊢\phi \to B\in ℂ$$
4		cxpneg	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to A^{-B}=\frac{1}{A^B}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^{-B}=\frac{1}{A^B}$$