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Definition df-hmop 22370
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
StepHypRef Expression
1 cho 21476 . 2  class  HrmOp
2 vx . . . . . . . 8  set  x
32cv 1618 . . . . . . 7  class  x
4 vy . . . . . . . . 9  set  y
54cv 1618 . . . . . . . 8  class  y
6 vt . . . . . . . . 9  set  t
76cv 1618 . . . . . . . 8  class  t
85, 7cfv 4659 . . . . . . 7  class  ( t `
 y )
9 csp 21448 . . . . . . 7  class  .ih
103, 8, 9co 5778 . . . . . 6  class  ( x 
.ih  ( t `  y ) )
113, 7cfv 4659 . . . . . . 7  class  ( t `
 x )
1211, 5, 9co 5778 . . . . . 6  class  ( ( t `  x ) 
.ih  y )
1310, 12wceq 1619 . . . . 5  wff  ( x 
.ih  ( t `  y ) )  =  ( ( t `  x )  .ih  y
)
14 chil 21445 . . . . 5  class  ~H
1513, 4, 14wral 2516 . . . 4  wff  A. y  e.  ~H  ( x  .ih  ( t `  y
) )  =  ( ( t `  x
)  .ih  y )
1615, 2, 14wral 2516 . . 3  wff  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( t `  y
) )  =  ( ( t `  x
)  .ih  y )
17 cmap 6726 . . . 4  class  ^m
1814, 14, 17co 5778 . . 3  class  ( ~H 
^m  ~H )
1916, 6, 18crab 2520 . 2  class  { t  e.  ( ~H  ^m  ~H )  |  A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( t `  y
) )  =  ( ( t `  x
)  .ih  y ) }
201, 19wceq 1619 1  wff  HrmOp  =  {
t  e.  ( ~H 
^m  ~H )  |  A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( t `  y
) )  =  ( ( t `  x
)  .ih  y ) }
Colors of variables: wff set class
This definition is referenced by:  elhmop  22399
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