Metamath Proof Explorer

Theorem lt0neg1d

Description: Comparison of a number and its negative to zero. Theorem I.23 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypothesis leidd.1 
|- ( ph -> A e. RR )
Assertion lt0neg1d 
|- ( ph -> ( A < 0 <-> 0 < -u A ) )

Proof

Step Hyp Ref Expression
1 leidd.1 
 |-  ( ph -> A e. RR )
2 lt0neg1 
 |-  ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
3 1 2 syl 
 |-  ( ph -> ( A < 0 <-> 0 < -u A ) )