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Metamath Proof Explorer

Theorem nn0risefaccl

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

		Ref	Expression
	Assertion	nn0risefaccl	$⊢ (A \in ℕ_{0} \land N \in ℕ_{0}) \to A^{\overline{N}} \in ℕ_{0}$

Step	Hyp	Ref	Expression
1		nn0sscn	$⊢ ℕ_{0} \subseteq ℂ$
2		1nn0	$⊢ 1 \in ℕ_{0}$
3		nn0mulcl	$⊢ (x \in ℕ_{0} \land y \in ℕ_{0}) \to x y \in ℕ_{0}$
4		nn0addcl	$⊢ (A \in ℕ_{0} \land k \in ℕ_{0}) \to A + k \in ℕ_{0}$
5	1 2 3 4	risefaccllem	$⊢ (A \in ℕ_{0} \land N \in ℕ_{0}) \to A^{\overline{N}} \in ℕ_{0}$