Theorem absmuli

Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of Gleason p. 133. (Contributed by NM, 1-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	$$⊢A\in ℂ$$
	abssub.2	$$⊢B\in ℂ$$
Assertion	absmuli	$$⊢\left|AB\right|=\left|A\right|\left|B\right|$$

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	$$⊢A\in ℂ$$
2		abssub.2	$$⊢B\in ℂ$$
3		absmul	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left|AB\right|=\left|A\right|\left|B\right|$$
4	1 2 3	mp2an	$$⊢\left|AB\right|=\left|A\right|\left|B\right|$$