# 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 ) )`
` |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( ( bra ` A ) ` C ) = ( C .ih A ) )`
` |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( B x. ( ( bra ` A ) ` C ) ) = ( B x. ( C .ih A ) ) )`
` |-  ( ( A e. ~H /\ B e. CC /\ C e. ~H ) -> ( ( bra ` A ) ` ( B .h C ) ) = ( B x. ( ( bra ` A ) ` C ) ) )`