Theorem absdivd

Description: Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	$$⊢\phi \to A\in ℂ$$
	abssubd.2	$$⊢\phi \to B\in ℂ$$
	absdivd.2	$$⊢\phi \to B\ne 0$$
Assertion	absdivd	$$⊢\phi \to \left|\frac{A}{B}\right|=\frac{\left|A\right|}{\left|B\right|}$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		abssubd.2	$$⊢\phi \to B\in ℂ$$
3		absdivd.2	$$⊢\phi \to B\ne 0$$
4		absdiv	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left|\frac{A}{B}\right|=\frac{\left|A\right|}{\left|B\right|}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left|\frac{A}{B}\right|=\frac{\left|A\right|}{\left|B\right|}$$