Theorem zmodfzp1

Description: An integer mod B lies in the first B + 1 nonnegative integers. (Contributed by AV, 27-Oct-2018)

	Ref	Expression
Assertion	zmodfzp1	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 mod 𝐵 ) ∈ ( 0 ... 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fzossfz	⊢ ( 0 ..^ 𝐵 ) ⊆ ( 0 ... 𝐵 )
2		zmodfzo	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 mod 𝐵 ) ∈ ( 0 ..^ 𝐵 ) )
3	1 2	sseldi	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 mod 𝐵 ) ∈ ( 0 ... 𝐵 ) )