Metamath Proof Explorer

Theorem ltnegi

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 21-Jan-1997)

Ref Expression
Hypotheses lt2.1 
|- A e. RR
lt2.2 
|- B e. RR
Assertion ltnegi 
|- ( A < B <-> -u B < -u A )

Proof

Step Hyp Ref Expression
1 lt2.1 
 |-  A e. RR
2 lt2.2 
 |-  B e. RR
3 ltneg 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4 1 2 3 mp2an 
 |-  ( A < B <-> -u B < -u A )