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Metamath Proof Explorer

Theorem fallfaccl

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

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

Step	Hyp	Ref	Expression
1		ssid	$⊢ ℂ \subseteq ℂ$
2		ax-1cn	$⊢ 1 \in ℂ$
3		mulcl	$⊢ (x \in ℂ \land y \in ℂ) \to x y \in ℂ$
4		nn0cn	$⊢ k \in ℕ_{0} \to k \in ℂ$
5		subcl	$⊢ (A \in ℂ \land k \in ℂ) \to A - k \in ℂ$
6	4 5	sylan2	$⊢ (A \in ℂ \land k \in ℕ_{0}) \to A - k \in ℂ$
7	1 2 3 6	fallfaccllem	$⊢ (A \in ℂ \land N \in ℕ_{0}) \to A^{\underline{N}} \in ℂ$