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Metamath Proof Explorer

Theorem hfmmval

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

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

Proof

Step Hyp Ref Expression
1 cnex 
 |-  CC e. _V
2 ax-hilex 
 |-  ~H e. _V
3 1 2 elmap 
 |-  ( T e. ( CC ^m ~H ) <-> T : ~H --> CC )
4 oveq1 
 |-  ( f = A -> ( f x. ( g ` x ) ) = ( A x. ( g ` x ) ) )
5 4 mpteq2dv 
 |-  ( f = A -> ( x e. ~H |-> ( f x. ( g ` x ) ) ) = ( x e. ~H |-> ( A x. ( g ` x ) ) ) )
6 fveq1 
 |-  ( g = T -> ( g ` x ) = ( T ` x ) )
7 6 oveq2d 
 |-  ( g = T -> ( A x. ( g ` x ) ) = ( A x. ( T ` x ) ) )
8 7 mpteq2dv 
 |-  ( g = T -> ( x e. ~H |-> ( A x. ( g ` x ) ) ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )
9 df-hfmul 
 |-  .fn = ( f e. CC , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( f x. ( g ` x ) ) ) )
10 2 mptex 
 |-  ( x e. ~H |-> ( A x. ( T ` x ) ) ) e. _V
11 5 8 9 10 ovmpo 
 |-  ( ( A e. CC /\ T e. ( CC ^m ~H ) ) -> ( A .fn T ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )
12 3 11 sylan2br 
 |-  ( ( A e. CC /\ T : ~H --> CC ) -> ( A .fn T ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )