Theorem abstrid

Description: Triangle inequality for absolute value. Proposition 10-3.7(h) of Gleason p. 133. (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	abstrid	$$⊢\phi \to \left|A+B\right|\le \left|A\right|+\left|B\right|$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		abssubd.2	$$⊢\phi \to B\in ℂ$$
3		abstri	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left|A+B\right|\le \left|A\right|+\left|B\right|$$
4	1 2 3	syl2anc	$$⊢\phi \to \left|A+B\right|\le \left|A\right|+\left|B\right|$$