Theorem cxpcld

Description: Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	$$⊢\phi \to A\in ℂ$$
	cxpcld.2	$$⊢\phi \to B\in ℂ$$
Assertion	cxpcld	$$⊢\phi \to A^B\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	$$⊢\phi \to A\in ℂ$$
2		cxpcld.2	$$⊢\phi \to B\in ℂ$$
3		cxpcl	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A^B\in ℂ$$
4	1 2 3	syl2anc	$$⊢\phi \to A^B\in ℂ$$