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 ∈ ℂ
	abssub.2	⊢ 𝐵 ∈ ℂ
Assertion	abstrii	⊢ ( abs ‘ ( 𝐴 + 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	⊢ 𝐴 ∈ ℂ
2		abssub.2	⊢ 𝐵 ∈ ℂ
3		abstri	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 + 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( abs ‘ ( 𝐴 + 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) )