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Metamath Proof Explorer

Theorem hosmval

Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

Ref Expression
Assertion hosmval 
|- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )

Proof

Step Hyp Ref Expression
1 ax-hilex 
 |-  ~H e. _V
2 1 1 elmap 
 |-  ( S e. ( ~H ^m ~H ) <-> S : ~H --> ~H )
3 1 1 elmap 
 |-  ( T e. ( ~H ^m ~H ) <-> T : ~H --> ~H )
4 fveq1 
 |-  ( f = S -> ( f ` x ) = ( S ` x ) )
5 4 oveq1d 
 |-  ( f = S -> ( ( f ` x ) +h ( g ` x ) ) = ( ( S ` x ) +h ( g ` x ) ) )
6 5 mpteq2dv 
 |-  ( f = S -> ( x e. ~H |-> ( ( f ` x ) +h ( g ` x ) ) ) = ( x e. ~H |-> ( ( S ` x ) +h ( g ` x ) ) ) )
7 fveq1 
 |-  ( g = T -> ( g ` x ) = ( T ` x ) )
8 7 oveq2d 
 |-  ( g = T -> ( ( S ` x ) +h ( g ` x ) ) = ( ( S ` x ) +h ( T ` x ) ) )
9 8 mpteq2dv 
 |-  ( g = T -> ( x e. ~H |-> ( ( S ` x ) +h ( g ` x ) ) ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )
10 df-hosum 
 |-  +op = ( f e. ( ~H ^m ~H ) , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) +h ( g ` x ) ) ) )
11 1 mptex 
 |-  ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) e. _V
12 6 9 10 11 ovmpo 
 |-  ( ( S e. ( ~H ^m ~H ) /\ T e. ( ~H ^m ~H ) ) -> ( S +op T ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )
13 2 3 12 syl2anbr 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )