Theorem abs2difabsd

Description: Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	$$⊢\phi \to A\in ℂ$$
	abssubd.2	$$⊢\phi \to B\in ℂ$$
Assertion	abs2difabsd	$$⊢\phi \to \left|\left|A\right|-\left|B\right|\right|\le \left|A-B\right|$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		abssubd.2	$$⊢\phi \to B\in ℂ$$
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	syl2anc	$$⊢\phi \to \left|\left|A\right|-\left|B\right|\right|\le \left|A-B\right|$$