Theorem phicld

Description: Closure for the value of the Euler phi function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	phicld.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
Assertion	phicld	⊢ ( 𝜑 → ( ϕ ‘ 𝑁 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		phicld.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		phicl	⊢ ( 𝑁 ∈ ℕ → ( ϕ ‘ 𝑁 ) ∈ ℕ )
3	1 2	syl	⊢ ( 𝜑 → ( ϕ ‘ 𝑁 ) ∈ ℕ )