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Metamath Proof Explorer

Theorem homulcl

Description: The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

Ref Expression
Assertion homulcl 
|- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )

Proof

Step Hyp Ref Expression
1 ffvelcdm 
 |-  ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
2 hvmulcl 
 |-  ( ( A e. CC /\ ( T ` x ) e. ~H ) -> ( A .h ( T ` x ) ) e. ~H )
3 1 2 sylan2 
 |-  ( ( A e. CC /\ ( T : ~H --> ~H /\ x e. ~H ) ) -> ( A .h ( T ` x ) ) e. ~H )
4 3 anassrs 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( T ` x ) ) e. ~H )
5 4 fmpttd 
 |-  ( ( A e. CC /\ T : ~H --> ~H ) -> ( x e. ~H |-> ( A .h ( T ` x ) ) ) : ~H --> ~H )
6 hommval 
 |-  ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )
7 6 feq1d 
 |-  ( ( A e. CC /\ T : ~H --> ~H ) -> ( ( A .op T ) : ~H --> ~H <-> ( x e. ~H |-> ( A .h ( T ` x ) ) ) : ~H --> ~H ) )
8 5 7 mpbird 
 |-  ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )