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Metamath Proof Explorer

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 )