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Metamath Proof Explorer

Theorem hoaddcl

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
Assertion hoaddcl 
|- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) : ~H --> ~H )

Proof

Step Hyp Ref Expression
1 ffvelcdm 
 |-  ( ( S : ~H --> ~H /\ x e. ~H ) -> ( S ` x ) e. ~H )
2 1 adantlr 
 |-  ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( S ` x ) e. ~H )
3 ffvelcdm 
 |-  ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
4 3 adantll 
 |-  ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( T ` x ) e. ~H )
5 hvaddcl 
 |-  ( ( ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( S ` x ) +h ( T ` x ) ) e. ~H )
6 2 4 5 syl2anc 
 |-  ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( S ` x ) +h ( T ` x ) ) e. ~H )
7 6 fmpttd 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) : ~H --> ~H )
8 hosmval 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )
9 8 feq1d 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( ( S +op T ) : ~H --> ~H <-> ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) : ~H --> ~H ) )
10 7 9 mpbird 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) : ~H --> ~H )