Definition df-hmop 9710

Description: Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators," sometimes with slightly different technical meanings.

Assertion

Ref	Expression
df-hmop	⊢ HrmOp = {t∣(t: ℋ –→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ (x ·ih (t ‘y)) = ((t ‘x) ·ih y))}

Distinct variable group:   x,t,y

Detailed syntax breakdown of Definition df-hmop

Step	Hyp	Ref	Expression
1		cho 8758	. 2 class HrmOp
2		chil 8727	. . . . 5 class ℋ
3		vt	. . . . . 6 set t
4	3	cv 953	. . . . 5 class t
5	2, 2, 4	wf 3173	. . . 4 wff t: ℋ –→ ℋ
6		vx	. . . . . . . . 9 set x
7	6	cv 953	. . . . . . . 8 class x
8		vy	. . . . . . . . . 10 set y
9	8	cv 953	. . . . . . . . 9 class y
10	9, 4	cfv 3177	. . . . . . . 8 class (t ‘y)
11		csp 8732	. . . . . . . 8 class ·ih
12	7, 10, 11	co 3954	. . . . . . 7 class (x ·ih (t ‘y))
13	7, 4	cfv 3177	. . . . . . . 8 class (t ‘x)
14	13, 9, 11	co 3954	. . . . . . 7 class ((t ‘x) ·ih y)
15	12, 14	wceq 954	. . . . . 6 wff (x ·ih (t ‘y)) = ((t ‘x) ·ih y)
16	15, 8, 2	wral 1642	. . . . 5 wff ∀y ∈ ℋ (x ·ih (t ‘y)) = ((t ‘x) ·ih y)
17	16, 6, 2	wral 1642	. . . 4 wff ∀x ∈ ℋ ∀y ∈ ℋ (x ·ih (t ‘y)) = ((t ‘x) ·ih y)
18	5, 17	wa 223	. . 3 wff (t: ℋ –→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ (x ·ih (t ‘y)) = ((t ‘x) ·ih y))
19	18, 3	cab 1461	. 2 class {t∣(t: ℋ –→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ (x ·ih (t ‘y)) = ((t ‘x) ·ih y))}
20	1, 19	wceq 954	1 wff HrmOp = {t∣(t: ℋ –→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ (x ·ih (t ‘y)) = ((t ‘x) ·ih y))}

This definition is referenced by:  elhmopt 9740

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