Step |
Hyp |
Ref |
Expression |
1 |
|
recn |
|- ( A e. RR -> A e. CC ) |
2 |
|
recn |
|- ( B e. RR -> B e. CC ) |
3 |
|
mulcom |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) |
4 |
1 2 3
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) = ( B x. A ) ) |
5 |
4
|
breq2d |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) <-> 0 < ( B x. A ) ) ) |
6 |
5
|
biimpd |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) -> 0 < ( B x. A ) ) ) |
7 |
|
prodgt0 |
|- ( ( ( B e. RR /\ A e. RR ) /\ ( 0 <_ B /\ 0 < ( B x. A ) ) ) -> 0 < A ) |
8 |
7
|
ex |
|- ( ( B e. RR /\ A e. RR ) -> ( ( 0 <_ B /\ 0 < ( B x. A ) ) -> 0 < A ) ) |
9 |
8
|
ancoms |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <_ B /\ 0 < ( B x. A ) ) -> 0 < A ) ) |
10 |
6 9
|
sylan2d |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <_ B /\ 0 < ( A x. B ) ) -> 0 < A ) ) |
11 |
10
|
imp |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ B /\ 0 < ( A x. B ) ) ) -> 0 < A ) |