Step |
Hyp |
Ref |
Expression |
1 |
|
1re |
|- 1 e. RR |
2 |
|
ltmul2 |
|- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) ) |
3 |
1 2
|
mp3an1 |
|- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) ) |
4 |
3
|
3impb |
|- ( ( B e. RR /\ A e. RR /\ 0 < A ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) ) |
5 |
4
|
3com12 |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) ) |
6 |
|
ax-1rid |
|- ( A e. RR -> ( A x. 1 ) = A ) |
7 |
6
|
3ad2ant1 |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( A x. 1 ) = A ) |
8 |
7
|
breq1d |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( ( A x. 1 ) < ( A x. B ) <-> A < ( A x. B ) ) ) |
9 |
5 8
|
bitrd |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) ) |