Metamath Proof Explorer

Theorem leneg

Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion leneg 
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 lesub2 
 |-  ( ( A e. RR /\ B e. RR /\ 0 e. RR ) -> ( A <_ B <-> ( 0 - B ) <_ ( 0 - A ) ) )
3 1 2 mp3an3 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> ( 0 - B ) <_ ( 0 - A ) ) )
4 df-neg 
 |-  -u B = ( 0 - B )
5 df-neg 
 |-  -u A = ( 0 - A )
6 4 5 breq12i 
 |-  ( -u B <_ -u A <-> ( 0 - B ) <_ ( 0 - A ) )
7 3 6 bitr4di 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )