Step |
Hyp |
Ref |
Expression |
1 |
|
mulcom |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) |
2 |
1
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) ) |
3 |
2
|
3adant3 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) ) |
4 |
|
mulass |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) |
5 |
|
mulass |
|- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) ) |
6 |
5
|
3com12 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) ) |
7 |
3 4 6
|
3eqtr3d |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) ) |