Theorem nn0expcld

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

	Ref	Expression
Hypotheses	nn0expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℕ0 )
	nn0expcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	nn0expcld	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		nn0expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℕ0 )
2		nn0expcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3		nn0expcl	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℕ0 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℕ0 )