Definition df-lnop 10047

Description: Define the set of linear operators on Hilbert space. (See df-hosum 9782 for definition of operator.)

Assertion

Ref	Expression
df-lnop	|- LinOp = {t | (t:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)))}

Distinct variable group:   x,t,y,z

Detailed syntax breakdown of Definition df-lnop

Step	Hyp	Ref	Expression
1		clo 9091	. 2 class LinOp
2		chil 9063	. . . . 5 class H~
3		vt	. . . . . 6 set t
4	3	cv 991	. . . . 5 class t
5	2, 2, 4	wf 3259	. . . 4 wff t:H~-->H~
6		vx	. . . . . . . . . . . 12 set x
7	6	cv 991	. . . . . . . . . . 11 class x
8		vy	. . . . . . . . . . . 12 set y
9	8	cv 991	. . . . . . . . . . 11 class y
10		csm 9065	. . . . . . . . . . 11 class .h
11	7, 9, 10	co 4021	. . . . . . . . . 10 class (x .h y)
12		vz	. . . . . . . . . . 11 set z
13	12	cv 991	. . . . . . . . . 10 class z
14		cva 9064	. . . . . . . . . 10 class +h
15	11, 13, 14	co 4021	. . . . . . . . 9 class ((x .h y) +h z)
16	15, 4	cfv 3263	. . . . . . . 8 class (t` ((x .h y) +h z))
17	9, 4	cfv 3263	. . . . . . . . . 10 class (t` y)
18	7, 17, 10	co 4021	. . . . . . . . 9 class (x .h (t` y))
19	13, 4	cfv 3263	. . . . . . . . 9 class (t` z)
20	18, 19, 14	co 4021	. . . . . . . 8 class ((x .h (t` y)) +h (t` z))
21	16, 20	wceq 992	. . . . . . 7 wff (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z))
22	21, 12, 2	wral 1691	. . . . . 6 wff A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z))
23	22, 8, 2	wral 1691	. . . . 5 wff A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z))
24		cc 5386	. . . . 5 class CC
25	23, 6, 24	wral 1691	. . . 4 wff A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z))
26	5, 25	wa 221	. . 3 wff (t:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)))
27	26, 3	cab 1505	. 2 class {t | (t:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)))}
28	1, 27	wceq 992	1 wff LinOp = {t | (t:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)))}

This definition is referenced by:  ellnop 10064  hhlnoi 10106

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