Theorem zmodcld

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

	Ref	Expression
Hypotheses	zmodcld.1	$$⊢\phi \to A\in \mathbb{Z}$$
	zmodcld.2	$$⊢\phi \to B\in \mathbb{N}$$
Assertion	zmodcld	$$⊢\phi \to A\mathrm{mod}B\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		zmodcld.1	$$⊢\phi \to A\in \mathbb{Z}$$
2		zmodcld.2	$$⊢\phi \to B\in \mathbb{N}$$
3		zmodcl	$$⊢\left(A\in \mathbb{Z}\wedge B\in \mathbb{N}\right)\to A\mathrm{mod}B\in {\mathbb{N}}_0$$
4	1 2 3	syl2anc	$$⊢\phi \to A\mathrm{mod}B\in {\mathbb{N}}_0$$