Metamath Proof Explorer


Theorem ltnegcon1

Description: Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004)

Ref Expression
Assertion ltnegcon1
|- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )

Proof

Step Hyp Ref Expression
1 renegcl
 |-  ( A e. RR -> -u A e. RR )
2 ltneg
 |-  ( ( -u A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < -u -u A ) )
3 1 2 sylan
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < -u -u A ) )
4 simpl
 |-  ( ( A e. RR /\ B e. RR ) -> A e. RR )
5 4 recnd
 |-  ( ( A e. RR /\ B e. RR ) -> A e. CC )
6 5 negnegd
 |-  ( ( A e. RR /\ B e. RR ) -> -u -u A = A )
7 6 breq2d
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u B < -u -u A <-> -u B < A ) )
8 3 7 bitrd
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )