Metamath Proof Explorer

Theorem xle0neg1

Description: Extended real version of le0neg1 . (Contributed by Mario Carneiro, 9-Sep-2015)

Ref Expression
Assertion xle0neg1 
|- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )

Proof

Step Hyp Ref Expression
1 0xr 
 |-  0 e. RR*
2 xleneg 
 |-  ( ( A e. RR* /\ 0 e. RR* ) -> ( A <_ 0 <-> -e 0 <_ -e A ) )
3 1 2 mpan2 
 |-  ( A e. RR* -> ( A <_ 0 <-> -e 0 <_ -e A ) )
4 xneg0 
 |-  -e 0 = 0
5 4 breq1i 
 |-  ( -e 0 <_ -e A <-> 0 <_ -e A )
6 3 5 bitrdi 
 |-  ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )