Definition df-hfsum 28592

Description: Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from ~H to CC, not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-hfsum	|- +fn = ( f e. ( CC ^m ~H ) , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) ) )

Distinct variable group:   f, g, x

Detailed syntax breakdown of Definition df-hfsum

Step	Hyp	Ref	Expression
1		chfs 27798	. 2 class +fn
2		vf	. . 3 setvar f
3		vg	. . 3 setvar g
4		cc 9934	. . . 4 class CC
5		chil 27776	. . . 4 class ~H
6		cmap 7857	. . . 4 class ^m
7	4, 5, 6	co 6650	. . 3 class ( CC ^m ~H )
8		vx	. . . 4 setvar x
9	8	cv 1482	. . . . . 6 class x
10	2	cv 1482	. . . . . 6 class f
11	9, 10	cfv 5888	. . . . 5 class ( f ` x )
12	3	cv 1482	. . . . . 6 class g
13	9, 12	cfv 5888	. . . . 5 class ( g ` x )
14		caddc 9939	. . . . 5 class +
15	11, 13, 14	co 6650	. . . 4 class ( ( f ` x ) + ( g ` x ) )
16	8, 5, 15	cmpt 4729	. . 3 class ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) )
17	2, 3, 7, 7, 16	cmpt2 6652	. 2 class ( f e. ( CC ^m ~H ) , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) ) )
18	1, 17	wceq 1483	1 wff +fn = ( f e. ( CC ^m ~H ) , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) + ( g ` x ) ) ) )

This definition is referenced by:  hfsmval  28597