Theorem expcld

Description: Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	expcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	expcld	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		expcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3		expcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )