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 ) ) |
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 ) ) |