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Metamath Proof Explorer

Theorem hommval

Description: Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

Ref Expression
Assertion hommval 
|- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )

Proof

Step Hyp Ref Expression
1 ax-hilex 
 |-  ~H e. _V
2 1 1 elmap 
 |-  ( T e. ( ~H ^m ~H ) <-> T : ~H --> ~H )
3 oveq1 
 |-  ( f = A -> ( f .h ( g ` x ) ) = ( A .h ( g ` x ) ) )
4 3 mpteq2dv 
 |-  ( f = A -> ( x e. ~H |-> ( f .h ( g ` x ) ) ) = ( x e. ~H |-> ( A .h ( g ` x ) ) ) )
5 fveq1 
 |-  ( g = T -> ( g ` x ) = ( T ` x ) )
6 5 oveq2d 
 |-  ( g = T -> ( A .h ( g ` x ) ) = ( A .h ( T ` x ) ) )
7 6 mpteq2dv 
 |-  ( g = T -> ( x e. ~H |-> ( A .h ( g ` x ) ) ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )
8 df-homul 
 |-  .op = ( f e. CC , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( f .h ( g ` x ) ) ) )
9 1 mptex 
 |-  ( x e. ~H |-> ( A .h ( T ` x ) ) ) e. _V
10 4 7 8 9 ovmpo 
 |-  ( ( A e. CC /\ T e. ( ~H ^m ~H ) ) -> ( A .op T ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )
11 2 10 sylan2br 
 |-  ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )