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Metamath Proof Explorer

Theorem homul12

Description: Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion homul12 
|- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( A .op ( B .op T ) ) = ( B .op ( A .op T ) ) )

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 ) .op T ) = ( ( B x. A ) .op T ) )
3 2 3adant3 
 |-  ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A x. B ) .op T ) = ( ( B x. A ) .op T ) )
4 homulass 
 |-  ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A x. B ) .op T ) = ( A .op ( B .op T ) ) )
5 homulass 
 |-  ( ( B e. CC /\ A e. CC /\ T : ~H --> ~H ) -> ( ( B x. A ) .op T ) = ( B .op ( A .op T ) ) )
6 5 3com12 
 |-  ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( B x. A ) .op T ) = ( B .op ( A .op T ) ) )
7 3 4 6 3eqtr3d 
 |-  ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( A .op ( B .op T ) ) = ( B .op ( A .op T ) ) )