| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simp1l |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> A e. RR ) |
| 2 |
1 1
|
remulcld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( A x. A ) e. RR ) |
| 3 |
|
simp2 |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> B e. RR ) |
| 4 |
3 1
|
remulcld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( B x. A ) e. RR ) |
| 5 |
3 3
|
remulcld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( B x. B ) e. RR ) |
| 6 |
|
simp1 |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( A e. RR /\ 0 <_ A ) ) |
| 7 |
|
simp3 |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> A < B ) |
| 8 |
1 3 7
|
ltled |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> A <_ B ) |
| 9 |
|
lemul1a |
|- ( ( ( A e. RR /\ B e. RR /\ ( A e. RR /\ 0 <_ A ) ) /\ A <_ B ) -> ( A x. A ) <_ ( B x. A ) ) |
| 10 |
1 3 6 8 9
|
syl31anc |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( A x. A ) <_ ( B x. A ) ) |
| 11 |
|
0red |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> 0 e. RR ) |
| 12 |
|
simp1r |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> 0 <_ A ) |
| 13 |
11 1 3 12 7
|
lelttrd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> 0 < B ) |
| 14 |
|
ltmul2 |
|- ( ( A e. RR /\ B e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( B x. A ) < ( B x. B ) ) ) |
| 15 |
1 3 3 13 14
|
syl112anc |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( A < B <-> ( B x. A ) < ( B x. B ) ) ) |
| 16 |
7 15
|
mpbid |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( B x. A ) < ( B x. B ) ) |
| 17 |
2 4 5 10 16
|
lelttrd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( A x. A ) < ( B x. B ) ) |