Metamath Proof Explorer


Theorem abssubge0d

Description: Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses absltd.1
|- ( ph -> A e. RR )
absltd.2
|- ( ph -> B e. RR )
abssubge0d.2
|- ( ph -> A <_ B )
Assertion abssubge0d
|- ( ph -> ( abs ` ( B - A ) ) = ( B - A ) )

Proof

Step Hyp Ref Expression
1 absltd.1
 |-  ( ph -> A e. RR )
2 absltd.2
 |-  ( ph -> B e. RR )
3 abssubge0d.2
 |-  ( ph -> A <_ B )
4 abssubge0
 |-  ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( abs ` ( B - A ) ) = ( B - A ) )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( abs ` ( B - A ) ) = ( B - A ) )