Definition df-homul 10932

Description: Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111.

Assertion

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

Distinct variable group:   f,g,h,x,y

Detailed syntax breakdown of Definition df-homul

Step	Hyp	Ref	Expression
1		chot 10232	. 2 class .op
2		vf	. . . . . . 7 set f
3	2	cv 1135	. . . . . 6 class f
4		cc 6180	. . . . . 6 class CC
5	3, 4	wcel 1138	. . . . 5 wff f e. CC
6		chil 10212	. . . . . 6 class ~H
7		vg	. . . . . . 7 set g
8	7	cv 1135	. . . . . 6 class g
9	6, 6, 8	wf 3805	. . . . 5 wff g:~H-->~H
10	5, 9	wa 239	. . . 4 wff (f e. CC /\ g:~H-->~H)
11		vh	. . . . . 6 set h
12	11	cv 1135	. . . . 5 class h
13		vx	. . . . . . . . 9 set x
14	13	cv 1135	. . . . . . . 8 class x
15	14, 6	wcel 1138	. . . . . . 7 wff x e. ~H
16		vy	. . . . . . . . 9 set y
17	16	cv 1135	. . . . . . . 8 class y
18	14, 8	cfv 3809	. . . . . . . . 9 class (g` x)
19		csm 10214	. . . . . . . . 9 class .h
20	3, 18, 19	co 4695	. . . . . . . 8 class (f .h (g` x))
21	17, 20	wceq 1136	. . . . . . 7 wff y = (f .h (g` x))
22	15, 21	wa 239	. . . . . 6 wff (x e. ~H /\ y = (f .h (g` x)))
23	22, 13, 16	copab 3213	. . . . 5 class {<.x, y>. | (x e. ~H /\ y = (f .h (g` x)))}
24	12, 23	wceq 1136	. . . 4 wff h = {<.x, y>. | (x e. ~H /\ y = (f .h (g` x)))}
25	10, 24	wa 239	. . 3 wff ((f e. CC /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = (f .h (g` x)))})
26	25, 2, 7, 11	copab2 4696	. 2 class {<.<.f, g>., h>. | ((f e. CC /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = (f .h (g` x)))})}
27	1, 26	wceq 1136	1 wff .op = {<.<.f, g>., h>. | ((f e. CC /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = (f .h (g` x)))})}

This definition is referenced by:  hommval 10937

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