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 ) ) ) )