Theorem cxprecd

Description: Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	cxprecd.2	$$⊢\phi \to B\in ℂ$$
Assertion	cxprecd	$$⊢\phi \to {\left(\frac{1}{A}\right)}^B=\frac{1}{A^B}$$

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		cxprecd.2	$$⊢\phi \to B\in ℂ$$
3		cxprec	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℂ\right)\to {\left(\frac{1}{A}\right)}^B=\frac{1}{A^B}$$
4	1 2 3	syl2anc	$$⊢\phi \to {\left(\frac{1}{A}\right)}^B=\frac{1}{A^B}$$