Metamath Proof Explorer


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 ) )