Theorem nn0risefaccl

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

	Ref	Expression
Assertion	nn0risefaccl	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 RiseFac 𝑁 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		nn0sscn	⊢ ℕ0 ⊆ ℂ
2		1nn0	⊢ 1 ∈ ℕ0
3		nn0mulcl	⊢ ( ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 · 𝑦 ) ∈ ℕ0 )
4		nn0addcl	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 + 𝑘 ) ∈ ℕ0 )
5	1 2 3 4	risefaccllem	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 RiseFac 𝑁 ) ∈ ℕ0 )