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Metamath Proof Explorer

Theorem bramul

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

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

Proof

Step Hyp Ref Expression
1 ax-his3 
 |-  ( ( B e. CC /\ C e. ~H /\ A e. ~H ) -> ( ( B .h C ) .ih A ) = ( B x. ( C .ih A ) ) )
2 1 3comr 
 |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( ( B .h C ) .ih A ) = ( B x. ( C .ih A ) ) )
3 hvmulcl 
 |-  ( ( B e. CC /\ 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. CC /\ C e. ~H ) ) -> ( ( bra ` A ) ` ( B .h C ) ) = ( ( B .h C ) .ih A ) )
6 5 3impb 
 |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( ( bra ` A ) ` ( B .h C ) ) = ( ( B .h C ) .ih A ) )
7 braval 
 |-  ( ( A e. ~H /\ C e. ~H ) -> ( ( bra ` A ) ` C ) = ( C .ih A ) )
8 7 3adant2 
 |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( ( bra ` A ) ` C ) = ( C .ih A ) )
9 8 oveq2d 
 |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( B x. ( ( bra ` A ) ` C ) ) = ( B x. ( C .ih A ) ) )
10 2 6 9 3eqtr4d 
 |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( ( bra ` A ) ` ( B .h C ) ) = ( B x. ( ( bra ` A ) ` C ) ) )