Step |
Hyp |
Ref |
Expression |
1 |
|
ltmulgt11 |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) ) |
2 |
|
recn |
|- ( A e. RR -> A e. CC ) |
3 |
|
recn |
|- ( B e. RR -> B e. CC ) |
4 |
|
mulcom |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) |
5 |
2 3 4
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) = ( B x. A ) ) |
6 |
5
|
3adant3 |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( A x. B ) = ( B x. A ) ) |
7 |
6
|
breq2d |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( A < ( A x. B ) <-> A < ( B x. A ) ) ) |
8 |
1 7
|
bitrd |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( B x. A ) ) ) |