Definition df-hodif 9425

Description: Define the difference of two Hilbert space operators. Definition of [Beran] p. 111.

Assertion

Ref	Expression
df-hodif	|- -op = {<.<.f, g>., h>. | ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) -h (g` x)))})}

Distinct variable group:   f,g,h,x,y

Detailed syntax breakdown of Definition df-hodif

Step	Hyp	Ref	Expression
1		chod 8748	. 2 class -op
2		chil 8727	. . . . . 6 class H~
3		vf	. . . . . . 7 set f
4	3	cv 952	. . . . . 6 class f
5	2, 2, 4	wf 3168	. . . . 5 wff f:H~-->H~
6		vg	. . . . . . 7 set g
7	6	cv 952	. . . . . 6 class g
8	2, 2, 7	wf 3168	. . . . 5 wff g:H~-->H~
9	5, 8	wa 223	. . . 4 wff (f:H~-->H~ /\ g:H~-->H~)
10		vh	. . . . . 6 set h
11	10	cv 952	. . . . 5 class h
12		vx	. . . . . . . . 9 set x
13	12	cv 952	. . . . . . . 8 class x
14	13, 2	wcel 955	. . . . . . 7 wff x e. H~
15		vy	. . . . . . . . 9 set y
16	15	cv 952	. . . . . . . 8 class y
17	13, 4	cfv 3172	. . . . . . . . 9 class (f` x)
18	13, 7	cfv 3172	. . . . . . . . 9 class (g` x)
19		cmv 8731	. . . . . . . . 9 class -h
20	17, 18, 19	co 3948	. . . . . . . 8 class ((f` x) -h (g` x))
21	16, 20	wceq 953	. . . . . . 7 wff y = ((f` x) -h (g` x))
22	14, 21	wa 223	. . . . . 6 wff (x e. H~ /\ y = ((f` x) -h (g` x)))
23	22, 12, 15	copab 2656	. . . . 5 class {<.x, y>. | (x e. H~ /\ y = ((f` x) -h (g` x)))}
24	11, 23	wceq 953	. . . 4 wff h = {<.x, y>. | (x e. H~ /\ y = ((f` x) -h (g` x)))}
25	9, 24	wa 223	. . 3 wff ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) -h (g` x)))})
26	25, 3, 6, 10	copab2 3949	. 2 class {<.<.f, g>., h>. | ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) -h (g` x)))})}
27	1, 26	wceq 953	1 wff -op = {<.<.f, g>., h>. | ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) -h (g` x)))})}

This definition is referenced by:  hodmvalt 9430

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