Theorem fznn0sub

Description: Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	fznn0sub	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑁 − 𝐾 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		elfzuz3	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) )
2		uznn0sub	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑁 − 𝐾 ) ∈ ℕ0 )
3	1 2	syl	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑁 − 𝐾 ) ∈ ℕ0 )