Theorem abs2difabsi

Description: Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007)

	Ref	Expression
Hypotheses	abs2difabsi.1	$$⊢A\in ℂ$$
	abs2difabsi.2	$$⊢B\in ℂ$$
Assertion	abs2difabsi	$$⊢\left|\left|A\right|-\left|B\right|\right|\le \left|A-B\right|$$

Proof

Step	Hyp	Ref	Expression
1		abs2difabsi.1	$$⊢A\in ℂ$$
2		abs2difabsi.2	$$⊢B\in ℂ$$
3		abs2difabs	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left|\left|A\right|-\left|B\right|\right|\le \left|A-B\right|$$
4	1 2 3	mp2an	$$⊢\left|\left|A\right|-\left|B\right|\right|\le \left|A-B\right|$$