This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem absori

Description: The absolute value of a real number is either that number or its negative. (Contributed by NM, 30-Sep-1999)

Ref Expression
Hypothesis sqrtthi.1 ⊢ 𝐴 ∈ ℝ
Assertion absori ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 )

Proof

Step Hyp Ref Expression
1 sqrtthi.1 ⊢ 𝐴 ∈ ℝ
2 absor ⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) )
3 1 2 ax-mp ⊢ ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 )