Metamath Proof Explorer


Theorem absord

Description: The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis resqrcld.1
|- ( ph -> A e. RR )
Assertion absord
|- ( ph -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1
 |-  ( ph -> A e. RR )
2 absor
 |-  ( A e. RR -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )
3 1 2 syl
 |-  ( ph -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )