Theorem kbval

Description: The value of the operator resulting from the outer product | A >. <. B | of two vectors. Equation 8.1 of Prugovecki p. 376. (Contributed by NM, 15-May-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	kbval	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( C .ih B ) .h A ) )

Proof

Step	Hyp	Ref	Expression
1		kbfval	|- ( ( A e. ~H /\ B e. ~H ) -> ( A ketbra B ) = ( x e. ~H |-> ( ( x .ih B ) .h A ) ) )
2	1	fveq1d	|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( x e. ~H |-> ( ( x .ih B ) .h A ) ) ` C ) )
3		oveq1	|- ( x = C -> ( x .ih B ) = ( C .ih B ) )
4	3	oveq1d	|- ( x = C -> ( ( x .ih B ) .h A ) = ( ( C .ih B ) .h A ) )
5		eqid	|- ( x e. ~H |-> ( ( x .ih B ) .h A ) ) = ( x e. ~H |-> ( ( x .ih B ) .h A ) )
6		ovex	|- ( ( C .ih B ) .h A ) e. _V
7	4 5 6	fvmpt	|- ( C e. ~H -> ( ( x e. ~H |-> ( ( x .ih B ) .h A ) ) ` C ) = ( ( C .ih B ) .h A ) )
8	2 7	sylan9eq	|- ( ( ( A e. ~H /\ B e. ~H ) /\ C e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( C .ih B ) .h A ) )
9	8	3impa	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( C .ih B ) .h A ) )