Metamath Proof Explorer

Theorem ltaddneg

Description: Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion ltaddneg 
|- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 ltadd2 
 |-  ( ( A e. RR /\ 0 e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < ( B + 0 ) ) )
3 1 2 mp3an2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < ( B + 0 ) ) )
4 recn 
 |-  ( B e. RR -> B e. CC )
5 4 addid1d 
 |-  ( B e. RR -> ( B + 0 ) = B )
6 5 adantl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B + 0 ) = B )
7 6 breq2d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) < ( B + 0 ) <-> ( B + A ) < B ) )
8 3 7 bitrd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) )