Metamath Proof Explorer


Theorem ltneg

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 27-Aug-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

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

Proof

Step Hyp Ref Expression
1 0re
 |-  0 e. RR
2 ltsub2
 |-  ( ( 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 ) )