Theorem zmodcld

Description: Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	zmodcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
	zmodcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℕ )
Assertion	zmodcld	⊢ ( 𝜑 → ( 𝐴 mod 𝐵 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		zmodcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
2		zmodcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℕ )
3		zmodcl	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 mod 𝐵 ) ∈ ℕ0 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 mod 𝐵 ) ∈ ℕ0 )