Metamath Proof Explorer


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 )