This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Database REAL AND COMPLEX NUMBERS Elementary real and complex functions Square root; absolute value 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 ) )