Theorem faccld

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

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

Proof

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