Metamath Proof Explorer

Theorem braadd

Description: Linearity property of bra for addition. (Contributed by NM, 23-May-2006) (New usage is discouraged.)

Ref Expression
Assertion braadd 
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` ( B +h C ) ) = ( ( ( bra ` A ) ` B ) + ( ( bra ` A ) ` C ) ) )

Proof

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 ) ) )