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Metamath Proof Explorer

Theorem hiassdi

Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005) (New usage is discouraged.)

Ref Expression
Assertion hiassdi 
|- ( ( ( 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 ) ) )

Proof

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 ) ) )