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Metamath Proof Explorer

Theorem hvmulcom

Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 mulcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
2 1 oveq1d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) .h C ) = ( ( B x. A ) .h C ) )
3 2 3adant3 
 |-  ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( ( B x. A ) .h C ) )
4 ax-hvmulass 
 |-  ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )
5 ax-hvmulass 
 |-  ( ( B e. CC /\ A e. CC /\ C e. ~H ) -> ( ( B x. A ) .h C ) = ( B .h ( A .h C ) ) )
6 5 3com12 
 |-  ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( B x. A ) .h C ) = ( B .h ( A .h C ) ) )
7 3 4 6 3eqtr3d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( A .h ( B .h C ) ) = ( B .h ( A .h C ) ) )