# Metamath Proof Explorer

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

Ref Expression
`|- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) : ~H --> ~H )`

### Proof

Step Hyp Ref Expression
1 ffvelrn
` |-  ( ( S : ~H --> ~H /\ x e. ~H ) -> ( S ` x ) e. ~H )`
` |-  ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( S ` x ) e. ~H )`
3 ffvelrn
` |-  ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )`
` |-  ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( T ` x ) e. ~H )`
` |-  ( ( ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( S ` x ) +h ( T ` x ) ) e. ~H )`
` |-  ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( S ` x ) +h ( T ` x ) ) e. ~H )`
` |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) : ~H --> ~H )`
` |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )`
` |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( ( S +op T ) : ~H --> ~H <-> ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) : ~H --> ~H ) )`
` |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) : ~H --> ~H )`