Theorem nnexpcld

Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

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