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

Metamath Proof Explorer

Theorem abssubge0

Description: Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008)

Ref Expression
Assertion abssubge0 
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( abs ` ( B - A ) ) = ( B - A ) )

Proof

Step Hyp Ref Expression
1 resubcl 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B - A ) e. RR )
2 1 3adant3 
 |-  ( ( B e. RR /\ A e. RR /\ A <_ B ) -> ( B - A ) e. RR )
3 subge0 
 |-  ( ( B e. RR /\ A e. RR ) -> ( 0 <_ ( B - A ) <-> A <_ B ) )
4 3 biimp3ar 
 |-  ( ( B e. RR /\ A e. RR /\ A <_ B ) -> 0 <_ ( B - A ) )
5 absid 
 |-  ( ( ( B - A ) e. RR /\ 0 <_ ( B - A ) ) -> ( abs ` ( B - A ) ) = ( B - A ) )
6 2 4 5 syl2anc 
 |-  ( ( B e. RR /\ A e. RR /\ A <_ B ) -> ( abs ` ( B - A ) ) = ( B - A ) )
7 6 3com12 
 |-  ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( abs ` ( B - A ) ) = ( B - A ) )