Definition df-hodif 28591

Description: Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   f, g, x

Detailed syntax breakdown of Definition df-hodif

Step	Hyp	Ref	Expression
1		chod 27797	. 2 class -op
2		vf	. . 3 setvar f
3		vg	. . 3 setvar g
4		chil 27776	. . . 4 class ~H
5		cmap 7857	. . . 4 class ^m
6	4, 4, 5	co 6650	. . 3 class ( ~H ^m ~H )
7		vx	. . . 4 setvar x
8	7	cv 1482	. . . . . 6 class x
9	2	cv 1482	. . . . . 6 class f
10	8, 9	cfv 5888	. . . . 5 class ( f ` x )
11	3	cv 1482	. . . . . 6 class g
12	8, 11	cfv 5888	. . . . 5 class ( g ` x )
13		cmv 27782	. . . . 5 class -h
14	10, 12, 13	co 6650	. . . 4 class ( ( f ` x ) -h ( g ` x ) )
15	7, 4, 14	cmpt 4729	. . 3 class ( x e. ~H |-> ( ( f ` x ) -h ( g ` x ) ) )
16	2, 3, 6, 6, 15	cmpt2 6652	. 2 class ( f e. ( ~H ^m ~H ) , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) -h ( g ` x ) ) ) )
17	1, 16	wceq 1483	1 wff -op = ( f e. ( ~H ^m ~H ) , g e. ( ~H ^m ~H ) |-> ( x e. ~H |-> ( ( f ` x ) -h ( g ` x ) ) ) )

This definition is referenced by:  hodmval  28596