Theorem abs3difd

Description: Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	$$⊢\phi \to A\in ℂ$$
	abssubd.2	$$⊢\phi \to B\in ℂ$$
	abs3difd.3	$$⊢\phi \to C\in ℂ$$
Assertion	abs3difd	$$⊢\phi \to \left|A-B\right|\le \left|A-C\right|+\left|C-B\right|$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		abssubd.2	$$⊢\phi \to B\in ℂ$$
3		abs3difd.3	$$⊢\phi \to C\in ℂ$$
4		abs3dif	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to \left|A-B\right|\le \left|A-C\right|+\left|C-B\right|$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left|A-B\right|\le \left|A-C\right|+\left|C-B\right|$$