This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nnexpcl

Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005)

		Ref	Expression
	Assertion	nnexpcl	$⊢ (A \in ℕ \land N \in ℕ_{0}) \to A^{N} \in ℕ$

Step	Hyp	Ref	Expression
1		nnsscn	$⊢ ℕ \subseteq ℂ$
2		nnmulcl	$⊢ (x \in ℕ \land y \in ℕ) \to x y \in ℕ$
3		1nn	$⊢ 1 \in ℕ$
4	1 2 3	expcllem	$⊢ (A \in ℕ \land N \in ℕ_{0}) \to A^{N} \in ℕ$