Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 27-Aug-1999) (Proof shortened by Mario Carneiro, 27-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ltneg | |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0re | |- 0 e. RR |
|
| 2 | ltsub2 | |- ( ( A e. RR /\ B e. RR /\ 0 e. RR ) -> ( A < B <-> ( 0 - B ) < ( 0 - A ) ) ) |
|
| 3 | 1 2 | mp3an3 | |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( 0 - B ) < ( 0 - A ) ) ) |
| 4 | df-neg | |- -u B = ( 0 - B ) |
|
| 5 | df-neg | |- -u A = ( 0 - A ) |
|
| 6 | 4 5 | breq12i | |- ( -u B < -u A <-> ( 0 - B ) < ( 0 - A ) ) |
| 7 | 3 6 | syl6bbr | |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) ) |