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Metamath Proof Explorer

Theorem hfsmval

Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

Ref Expression
Assertion hfsmval 
|- ( ( S : ~H --> CC /\ T : ~H --> CC ) -> ( S +fn T ) = ( x e. ~H |-> ( ( S ` x ) + ( T ` x ) ) ) )

Proof

Step Hyp Ref Expression
1 cnex 
 |-  CC e. _V
2 ax-hilex 
 |-  ~H e. _V
3 1 2 elmap 
 |-  ( S e. ( CC ^m ~H ) <-> S : ~H --> CC )
4 1 2 elmap 
 |-  ( T e. ( CC ^m ~H ) <-> T : ~H --> CC )
5 fveq1 
 |-  ( f = S -> ( f ` x ) = ( S ` x ) )
6 5 oveq1d 
 |-  ( f = S -> ( ( f ` x ) + ( g ` x ) ) = ( ( S ` x ) + ( g ` x ) ) )
7 6 mpteq2dv 
 |-  ( f = S -> ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) ) = ( x e. ~H |-> ( ( S ` x ) + ( g ` x ) ) ) )
8 fveq1 
 |-  ( g = T -> ( g ` x ) = ( T ` x ) )
9 8 oveq2d 
 |-  ( g = T -> ( ( S ` x ) + ( g ` x ) ) = ( ( S ` x ) + ( T ` x ) ) )
10 9 mpteq2dv 
 |-  ( g = T -> ( x e. ~H |-> ( ( S ` x ) + ( g ` x ) ) ) = ( x e. ~H |-> ( ( S ` x ) + ( T ` x ) ) ) )
11 df-hfsum 
 |-  +fn = ( f e. ( CC ^m ~H ) , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) ) )
12 2 mptex 
 |-  ( x e. ~H |-> ( ( S ` x ) + ( T ` x ) ) ) e. _V
13 7 10 11 12 ovmpo 
 |-  ( ( S e. ( CC ^m ~H ) /\ T e. ( CC ^m ~H ) ) -> ( S +fn T ) = ( x e. ~H |-> ( ( S ` x ) + ( T ` x ) ) ) )
14 3 4 13 syl2anbr 
 |-  ( ( S : ~H --> CC /\ T : ~H --> CC ) -> ( S +fn T ) = ( x e. ~H |-> ( ( S ` x ) + ( T ` x ) ) ) )