Theorem abssubi

Description: Swapping order of subtraction doesn't change the absolute value. Example of Apostol p. 363. (Contributed by NM, 1-Oct-1999)

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

Proof

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