Metamath Proof Explorer

Theorem xlt0neg2

Description: Extended real version of lt0neg2 . (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xlt0neg2 
|- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )

Proof

Step Hyp Ref Expression
1 0xr 
 |-  0 e. RR*
2 xltneg 
 |-  ( ( 0 e. RR* /\ A e. RR* ) -> ( 0 < A <-> -e A < -e 0 ) )
3 1 2 mpan 
 |-  ( A e. RR* -> ( 0 < A <-> -e A < -e 0 ) )
4 xneg0 
 |-  -e 0 = 0
5 4 breq2i 
 |-  ( -e A < -e 0 <-> -e A < 0 )
6 3 5 bitrdi 
 |-  ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )