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Metamath Proof Explorer

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 )