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Metamath Proof Explorer

Theorem nnrisefaccl

Description: Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018)

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

Step	Hyp	Ref	Expression
1		nnsscn	$⊢ ℕ \subseteq ℂ$
2		1nn	$⊢ 1 \in ℕ$
3		nnmulcl	$⊢ (x \in ℕ \land y \in ℕ) \to x y \in ℕ$
4		nnnn0addcl	$⊢ (A \in ℕ \land k \in ℕ_{0}) \to A + k \in ℕ$
5	1 2 3 4	risefaccllem	$⊢ (A \in ℕ \land N \in ℕ_{0}) \to A^{\overline{N}} \in ℕ$