Theorem abstrii

Description: Triangle inequality for absolute value. Proposition 10-3.7(h) of Gleason p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	$$⊢A\in ℂ$$
	abssub.2	$$⊢B\in ℂ$$
Assertion	abstrii	$$⊢\left|A+B\right|\le \left|A\right|+\left|B\right|$$

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	$$⊢A\in ℂ$$
2		abssub.2	$$⊢B\in ℂ$$
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	mp2an	$$⊢\left|A+B\right|\le \left|A\right|+\left|B\right|$$