Step |
Hyp |
Ref |
Expression |
1 |
|
renegcl |
|- ( A e. RR -> -u A e. RR ) |
2 |
|
ltneg |
|- ( ( -u A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < -u -u A ) ) |
3 |
1 2
|
sylan |
|- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < -u -u A ) ) |
4 |
|
simpl |
|- ( ( A e. RR /\ B e. RR ) -> A e. RR ) |
5 |
4
|
recnd |
|- ( ( A e. RR /\ B e. RR ) -> A e. CC ) |
6 |
5
|
negnegd |
|- ( ( A e. RR /\ B e. RR ) -> -u -u A = A ) |
7 |
6
|
breq2d |
|- ( ( A e. RR /\ B e. RR ) -> ( -u B < -u -u A <-> -u B < A ) ) |
8 |
3 7
|
bitrd |
|- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) ) |