Step |
Hyp |
Ref |
Expression |
1 |
|
1red |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 e. RR ) |
2 |
|
simpll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> A e. RR ) |
3 |
|
remulcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) |
4 |
3
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> ( A x. B ) e. RR ) |
5 |
|
simprl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < A ) |
6 |
|
simprr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < B ) |
7 |
|
0red |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 0 e. RR ) |
8 |
|
0lt1 |
|- 0 < 1 |
9 |
8
|
a1i |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 0 < 1 ) |
10 |
7 1 2 9 5
|
lttrd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 0 < A ) |
11 |
|
ltmulgt11 |
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) ) |
12 |
11
|
3expa |
|- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) ) |
13 |
10 12
|
syldan |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> ( 1 < B <-> A < ( A x. B ) ) ) |
14 |
6 13
|
mpbid |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> A < ( A x. B ) ) |
15 |
1 2 4 5 14
|
lttrd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) ) |