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Metamath Proof Explorer

Theorem rerisefaccl

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

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

Step	Hyp	Ref	Expression
1		ax-resscn	$⊢ ℝ \subseteq ℂ$
2		1re	$⊢ 1 \in ℝ$
3		remulcl	$⊢ (x \in ℝ \land y \in ℝ) \to x y \in ℝ$
4		nn0re	$⊢ k \in ℕ_{0} \to k \in ℝ$
5		readdcl	$⊢ (A \in ℝ \land k \in ℝ) \to A + k \in ℝ$
6	4 5	sylan2	$⊢ (A \in ℝ \land k \in ℕ_{0}) \to A + k \in ℝ$
7	1 2 3 6	risefaccllem	$⊢ (A \in ℝ \land N \in ℕ_{0}) \to A^{\overline{N}} \in ℝ$