Theorem sqabssubi

Description: Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007)

	Ref	Expression
Hypotheses	absvalsqi.1	$$⊢A\in ℂ$$
	abssub.2	$$⊢B\in ℂ$$
Assertion	sqabssubi	$$⊢{\left|A-B\right|}^2={\left|A\right|}^2+{\left|B\right|}^2-2\Re \left(A\bar{B}\right)$$

Proof

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