Metamath Proof Explorer

Theorem halfpos2

Description: A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005)

Ref Expression
Assertion halfpos2 
|- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )

Proof

Step Hyp Ref Expression
1 2re 
 |-  2 e. RR
2 2pos 
 |-  0 < 2
3 gt0div 
 |-  ( ( A e. RR /\ 2 e. RR /\ 0 < 2 ) -> ( 0 < A <-> 0 < ( A / 2 ) ) )
4 1 2 3 mp3an23 
 |-  ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )