Metamath Proof Explorer


Theorem absnidi

Description: A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999)

Ref Expression
Hypothesis sqrtthi.1
|- A e. RR
Assertion absnidi
|- ( A <_ 0 -> ( abs ` A ) = -u A )

Proof

Step Hyp Ref Expression
1 sqrtthi.1
 |-  A e. RR
2 absnid
 |-  ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A )
3 1 2 mpan
 |-  ( A <_ 0 -> ( abs ` A ) = -u A )