Metamath Proof Explorer

Theorem lt0neg1

Description: Comparison of a number and its negative to zero. Theorem I.23 of Apostol p. 20. (Contributed by NM, 14-May-1999)

Ref Expression
Assertion lt0neg1 
|- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 ltneg 
 |-  ( ( A e. RR /\ 0 e. RR ) -> ( A < 0 <-> -u 0 < -u A ) )
3 1 2 mpan2 
 |-  ( A e. RR -> ( A < 0 <-> -u 0 < -u A ) )
4 neg0 
 |-  -u 0 = 0
5 4 breq1i 
 |-  ( -u 0 < -u A <-> 0 < -u A )
6 3 5 syl6bb 
 |-  ( A e. RR -> ( A < 0 <-> 0 < -u A ) )