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Metamath Proof Explorer

Theorem ltnegcon2

Description: Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015)

Ref Expression
Assertion ltnegcon2 
|- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) )

Proof

Step Hyp Ref Expression
1 renegcl 
 |-  ( B e. RR -> -u B e. RR )
2 ltneg 
 |-  ( ( A e. RR /\ -u B e. RR ) -> ( A < -u B <-> -u -u B < -u A ) )
3 1 2 sylan2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> -u -u B < -u A ) )
4 simpr 
 |-  ( ( A e. RR /\ B e. RR ) -> B e. RR )
5 4 recnd 
 |-  ( ( A e. RR /\ B e. RR ) -> B e. CC )
6 5 negnegd 
 |-  ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
7 6 breq1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u -u B < -u A <-> B < -u A ) )
8 3 7 bitrd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) )