Step |
Hyp |
Ref |
Expression |
1 |
|
ax-his2 |
|- ( ( B e. ~H /\ C e. ~H /\ A e. ~H ) -> ( ( B +h C ) .ih A ) = ( ( B .ih A ) + ( C .ih A ) ) ) |
2 |
1
|
3comr |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( B +h C ) .ih A ) = ( ( B .ih A ) + ( C .ih A ) ) ) |
3 |
|
hvaddcl |
|- ( ( B e. ~H /\ C e. ~H ) -> ( B +h C ) e. ~H ) |
4 |
|
braval |
|- ( ( A e. ~H /\ ( B +h C ) e. ~H ) -> ( ( bra ` A ) ` ( B +h C ) ) = ( ( B +h C ) .ih A ) ) |
5 |
3 4
|
sylan2 |
|- ( ( A e. ~H /\ ( B e. ~H /\ C e. ~H ) ) -> ( ( bra ` A ) ` ( B +h C ) ) = ( ( B +h C ) .ih A ) ) |
6 |
5
|
3impb |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` ( B +h C ) ) = ( ( B +h C ) .ih A ) ) |
7 |
|
braval |
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( bra ` A ) ` B ) = ( B .ih A ) ) |
8 |
7
|
3adant3 |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` B ) = ( B .ih A ) ) |
9 |
|
braval |
|- ( ( A e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` C ) = ( C .ih A ) ) |
10 |
9
|
3adant2 |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` C ) = ( C .ih A ) ) |
11 |
8 10
|
oveq12d |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( bra ` A ) ` B ) + ( ( bra ` A ) ` C ) ) = ( ( B .ih A ) + ( C .ih A ) ) ) |
12 |
2 6 11
|
3eqtr4d |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` ( B +h C ) ) = ( ( ( bra ` A ) ` B ) + ( ( bra ` A ) ` C ) ) ) |