Theorem rpcxpcld

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

	Ref	Expression
Hypotheses	rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	rpcxpcld	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		rpcxpcl	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )