Theorem modcld

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

	Ref	Expression
Hypotheses	modcld.1	$$⊢\phi \to A\in ℝ$$
	modcld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	modcld	$$⊢\phi \to A\mathrm{mod}B\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		modcld.1	$$⊢\phi \to A\in ℝ$$
2		modcld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		modcl	$$⊢\left(A\in ℝ\wedge B\in {ℝ}^{+}\right)\to A\mathrm{mod}B\in ℝ$$
4	1 2 3	syl2anc	$$⊢\phi \to A\mathrm{mod}B\in ℝ$$