Theorem recxpcld

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

	Ref	Expression
Hypotheses	recxpcld.1	$$⊢\phi \to A\in ℝ$$
	recxpcld.2	$$⊢\phi \to 0\le A$$
	recxpcld.3	$$⊢\phi \to B\in ℝ$$
Assertion	recxpcld	$$⊢\phi \to A^B\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	$$⊢\phi \to A\in ℝ$$
2		recxpcld.2	$$⊢\phi \to 0\le A$$
3		recxpcld.3	$$⊢\phi \to B\in ℝ$$
4		recxpcl	$$⊢\left(A\in ℝ\wedge 0\le A\wedge B\in ℝ\right)\to A^B\in ℝ$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^B\in ℝ$$