Metamath Proof Explorer


Theorem sn-ltaddneg

Description: ltaddneg without ax-mulcom . (Contributed by SN, 25-Jan-2025)

Ref Expression
Assertion sn-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 readdrid
 |-  ( B e. RR -> ( B + 0 ) = B )
5 4 adantl
 |-  ( ( A e. RR /\ B e. RR ) -> ( B + 0 ) = B )
6 5 breq2d
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) < ( B + 0 ) <-> ( B + A ) < B ) )
7 3 6 bitrd
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) )