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Mirrors > Home > MPE Home > Th. List > nnexpcl | Structured version Visualization version GIF version |
Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
Ref | Expression |
---|---|
nnexpcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 11025 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | nnmulcl 11043 | . 2 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 · 𝑦) ∈ ℕ) | |
3 | 1nn 11031 | . 2 ⊢ 1 ∈ ℕ | |
4 | 1, 2, 3 | expcllem 12871 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) |
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