Theorem recxpcld

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

	Ref	Expression
Hypotheses	recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	recxpcld.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	recxpcld	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		recxpcld.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4		recxpcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ )