Theorem nnrisefaccl

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

	Ref	Expression
Assertion	nnrisefaccl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 RiseFac 𝑁 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		nnsscn	⊢ ℕ ⊆ ℂ
2		1nn	⊢ 1 ∈ ℕ
3		nnmulcl	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 · 𝑦 ) ∈ ℕ )
4		nnnn0addcl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 + 𝑘 ) ∈ ℕ )
5	1 2 3 4	risefaccllem	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 RiseFac 𝑁 ) ∈ ℕ )