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Metamath Proof Explorer

Theorem zfallfaccl

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

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

Step	Hyp	Ref	Expression
1		zsscn	$⊢ ℤ \subseteq ℂ$
2		1z	$⊢ 1 \in ℤ$
3		zmulcl	$⊢ (x \in ℤ \land y \in ℤ) \to x y \in ℤ$
4		nn0z	$⊢ k \in ℕ_{0} \to k \in ℤ$
5		zsubcl	$⊢ (A \in ℤ \land k \in ℤ) \to A - k \in ℤ$
6	4 5	sylan2	$⊢ (A \in ℤ \land k \in ℕ_{0}) \to A - k \in ℤ$
7	1 2 3 6	fallfaccllem	$⊢ (A \in ℤ \land N \in ℕ_{0}) \to A^{\underline{N}} \in ℤ$