Metamath Proof Explorer

Theorem his52

Description: Associative law for inner product. (Contributed by NM, 13-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion his52 
|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( B .ih ( ( * ` A ) .h C ) ) = ( A x. ( B .ih C ) ) )

Proof

Step Hyp Ref Expression
1 cjcl 
 |-  ( A e. CC -> ( * ` A ) e. CC )
2 his5 
 |-  ( ( ( * ` A ) e. CC /\ B e. ~H /\ C e. ~H ) -> ( B .ih ( ( * ` A ) .h C ) ) = ( ( * ` ( * ` A ) ) x. ( B .ih C ) ) )
3 1 2 syl3an1 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( B .ih ( ( * ` A ) .h C ) ) = ( ( * ` ( * ` A ) ) x. ( B .ih C ) ) )
4 cjcj 
 |-  ( A e. CC -> ( * ` ( * ` A ) ) = A )
5 4 oveq1d 
 |-  ( A e. CC -> ( ( * ` ( * ` A ) ) x. ( B .ih C ) ) = ( A x. ( B .ih C ) ) )
6 5 3ad2ant1 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( * ` ( * ` A ) ) x. ( B .ih C ) ) = ( A x. ( B .ih C ) ) )
7 3 6 eqtrd 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( B .ih ( ( * ` A ) .h C ) ) = ( A x. ( B .ih C ) ) )