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Metamath Proof Explorer

Theorem nn0expcl

Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005)

		Ref	Expression
	Assertion	nn0expcl	$⊢ (A \in ℕ_{0} \land N \in ℕ_{0}) \to A^{N} \in ℕ_{0}$

Step	Hyp	Ref	Expression
1		nn0sscn	$⊢ ℕ_{0} \subseteq ℂ$
2		nn0mulcl	$⊢ (x \in ℕ_{0} \land y \in ℕ_{0}) \to x y \in ℕ_{0}$
3		1nn0	$⊢ 1 \in ℕ_{0}$
4	1 2 3	expcllem	$⊢ (A \in ℕ_{0} \land N \in ℕ_{0}) \to A^{N} \in ℕ_{0}$