Theorem kbop

Description: The outer product of two vectors, expressed as | A >. <. B | in Dirac notation, is an operator. (Contributed by NM, 30-May-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	kbop	|- ( ( A e. ~H /\ B e. ~H ) -> ( A ketbra B ) : ~H --> ~H )

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		hicl	|- ( ( x e. ~H /\ B e. ~H ) -> ( x .ih B ) e. CC )
3		hvmulcl	|- ( ( ( x .ih B ) e. CC /\ A e. ~H ) -> ( ( x .ih B ) .h A ) e. ~H )
4	2 3	sylan	|- ( ( ( x e. ~H /\ B e. ~H ) /\ A e. ~H ) -> ( ( x .ih B ) .h A ) e. ~H )
5	4	an31s	|- ( ( ( A e. ~H /\ B e. ~H ) /\ x e. ~H ) -> ( ( x .ih B ) .h A ) e. ~H )
6	1 5	fmpt3d	|- ( ( A e. ~H /\ B e. ~H ) -> ( A ketbra B ) : ~H --> ~H )