Theorem rpcxpcld

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

	Ref	Expression
Hypotheses	rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpcxpcld.2	$$⊢\phi \to B\in ℝ$$
Assertion	rpcxpcld	$$⊢\phi \to A^B\in {ℝ}^{+}$$

Proof

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