Definition df-hmop 28703

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. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-hmop	|- HrmOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. ~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 27807	. 2 class HrmOp
2		vx	. . . . . . . 8 setvar x
3	2	cv 1482	. . . . . . 7 class x
4		vy	. . . . . . . . 9 setvar y
5	4	cv 1482	. . . . . . . 8 class y
6		vt	. . . . . . . . 9 setvar t
7	6	cv 1482	. . . . . . . 8 class t
8	5, 7	cfv 5888	. . . . . . 7 class ( t ` y )
9		csp 27779	. . . . . . 7 class .ih
10	3, 8, 9	co 6650	. . . . . 6 class ( x .ih ( t ` y ) )
11	3, 7	cfv 5888	. . . . . . 7 class ( t ` x )
12	11, 5, 9	co 6650	. . . . . 6 class ( ( t ` x ) .ih y )
13	10, 12	wceq 1483	. . . . 5 wff ( x .ih ( t ` y ) ) = ( ( t ` x ) .ih y )
14		chil 27776	. . . . 5 class ~H
15	13, 4, 14	wral 2912	. . . 4 wff A. y e. ~H ( x .ih ( t ` y ) ) = ( ( t ` x ) .ih y )
16	15, 2, 14	wral 2912	. . 3 wff A. x e. ~H A. y e. ~H ( x .ih ( t ` y ) ) = ( ( t ` x ) .ih y )
17		cmap 7857	. . . 4 class ^m
18	14, 14, 17	co 6650	. . 3 class ( ~H ^m ~H )
19	16, 6, 18	crab 2916	. 2 class { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. ~H ( x .ih ( t ` y ) ) = ( ( t ` x ) .ih y ) }
20	1, 19	wceq 1483	1 wff HrmOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. ~H ( x .ih ( t ` y ) ) = ( ( t ` x ) .ih y ) }

This definition is referenced by:  elhmop  28732