Step |
Hyp |
Ref |
Expression |
1 |
|
ldiv.a |
|- ( ph -> A e. CC ) |
2 |
|
ldiv.b |
|- ( ph -> B e. CC ) |
3 |
|
ldiv.c |
|- ( ph -> C e. CC ) |
4 |
|
ldiv.bn0 |
|- ( ph -> B =/= 0 ) |
5 |
|
oveq1 |
|- ( ( A x. B ) = C -> ( ( A x. B ) / B ) = ( C / B ) ) |
6 |
1 2 4
|
divcan4d |
|- ( ph -> ( ( A x. B ) / B ) = A ) |
7 |
6
|
eqeq1d |
|- ( ph -> ( ( ( A x. B ) / B ) = ( C / B ) <-> A = ( C / B ) ) ) |
8 |
5 7
|
syl5ib |
|- ( ph -> ( ( A x. B ) = C -> A = ( C / B ) ) ) |
9 |
|
oveq1 |
|- ( A = ( C / B ) -> ( A x. B ) = ( ( C / B ) x. B ) ) |
10 |
3 2 4
|
divcan1d |
|- ( ph -> ( ( C / B ) x. B ) = C ) |
11 |
10
|
eqeq2d |
|- ( ph -> ( ( A x. B ) = ( ( C / B ) x. B ) <-> ( A x. B ) = C ) ) |
12 |
9 11
|
syl5ib |
|- ( ph -> ( A = ( C / B ) -> ( A x. B ) = C ) ) |
13 |
8 12
|
impbid |
|- ( ph -> ( ( A x. B ) = C <-> A = ( C / B ) ) ) |