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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 ) )