Theorem absdivzi

Description: Absolute value distributes over division. (Contributed by NM, 26-Mar-2005)

	Ref	Expression
Hypotheses	absvalsqi.1	$$⊢A\in ℂ$$
	abssub.2	$$⊢B\in ℂ$$
Assertion	absdivzi	$$⊢B\ne 0\to \left|\frac{A}{B}\right|=\frac{\left|A\right|}{\left|B\right|}$$

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	$$⊢A\in ℂ$$
2		abssub.2	$$⊢B\in ℂ$$
3		absdiv	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left|\frac{A}{B}\right|=\frac{\left|A\right|}{\left|B\right|}$$
4	1 2 3	mp3an12	$$⊢B\ne 0\to \left|\frac{A}{B}\right|=\frac{\left|A\right|}{\left|B\right|}$$