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

Metamath Proof Explorer

Theorem expcl

Description: Closure law for nonnegative integer exponentiation. For integer exponents, see expclz . (Contributed by NM, 26-May-2005)

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

Step	Hyp	Ref	Expression
1		ssid	$⊢ ℂ \subseteq ℂ$
2		mulcl	$⊢ (x \in ℂ \land y \in ℂ) \to x y \in ℂ$
3		ax-1cn	$⊢ 1 \in ℂ$
4	1 2 3	expcllem	$⊢ (A \in ℂ \land N \in ℕ_{0}) \to A^{N} \in ℂ$