Step |
Hyp |
Ref |
Expression |
1 |
|
hvmulcl |
|- ( ( A e. CC /\ B e. ~H ) -> ( A .h B ) e. ~H ) |
2 |
|
ax-his2 |
|- ( ( ( A .h B ) e. ~H /\ C e. ~H /\ D e. ~H ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) ) |
3 |
2
|
3expb |
|- ( ( ( A .h B ) e. ~H /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) ) |
4 |
1 3
|
sylan |
|- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) ) |
5 |
|
ax-his3 |
|- ( ( A e. CC /\ B e. ~H /\ D e. ~H ) -> ( ( A .h B ) .ih D ) = ( A x. ( B .ih D ) ) ) |
6 |
5
|
3expa |
|- ( ( ( A e. CC /\ B e. ~H ) /\ D e. ~H ) -> ( ( A .h B ) .ih D ) = ( A x. ( B .ih D ) ) ) |
7 |
6
|
adantrl |
|- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A .h B ) .ih D ) = ( A x. ( B .ih D ) ) ) |
8 |
7
|
oveq1d |
|- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) = ( ( A x. ( B .ih D ) ) + ( C .ih D ) ) ) |
9 |
4 8
|
eqtrd |
|- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( A x. ( B .ih D ) ) + ( C .ih D ) ) ) |