Theorem cxpcld

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

	Ref	Expression
Hypotheses	cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	cxpcld	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		cxpcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )