Metamath Proof Explorer


Theorem faccld

Description: Closure of the factorial function, deduction version of faccl . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypothesis faccld.1 ( 𝜑𝑁 ∈ ℕ0 )
Assertion faccld ( 𝜑 → ( ! ‘ 𝑁 ) ∈ ℕ )

Proof

Step Hyp Ref Expression
1 faccld.1 ( 𝜑𝑁 ∈ ℕ0 )
2 faccl ( 𝑁 ∈ ℕ0 → ( ! ‘ 𝑁 ) ∈ ℕ )
3 1 2 syl ( 𝜑 → ( ! ‘ 𝑁 ) ∈ ℕ )