Definition df-hfmul 28593

Description: Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   f, g, x

Detailed syntax breakdown of Definition df-hfmul

Step	Hyp	Ref	Expression
1		chft 27799	. 2 class .fn
2		vf	. . 3 setvar f
3		vg	. . 3 setvar g
4		cc 9934	. . 3 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	2	cv 1482	. . . . 5 class f
10	8	cv 1482	. . . . . 6 class x
11	3	cv 1482	. . . . . 6 class g
12	10, 11	cfv 5888	. . . . 5 class ( g ` x )
13		cmul 9941	. . . . 5 class x.
14	9, 12, 13	co 6650	. . . 4 class ( f x. ( g ` x ) )
15	8, 5, 14	cmpt 4729	. . 3 class ( x e. ~H |-> ( f x. ( g ` x ) ) )
16	2, 3, 4, 7, 15	cmpt2 6652	. 2 class ( f e. CC , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( f x. ( g ` x ) ) ) )
17	1, 16	wceq 1483	1 wff .fn = ( f e. CC , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( f x. ( g ` x ) ) ) )

This definition is referenced by:  hfmmval  28598