Definition df-homul 28590

Description: Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   f, g, x

Detailed syntax breakdown of Definition df-homul

Step	Hyp	Ref	Expression
1		chot 27796	. 2 class .op
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	5, 5, 6	co 6650	. . 3 class ( ~H ^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		csm 27778	. . . . 5 class .h
14	9, 12, 13	co 6650	. . . 4 class ( f .h ( g ` x ) )
15	8, 5, 14	cmpt 4729	. . 3 class ( x e. ~H |-> ( f .h ( g ` x ) ) )
16	2, 3, 4, 7, 15	cmpt2 6652	. 2 class ( f e. CC , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( f .h ( g ` x ) ) ) )
17	1, 16	wceq 1483	1 wff .op = ( f e. CC , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( f .h ( g ` x ) ) ) )

This definition is referenced by:  hommval  28595