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Definition df-unop 22439
Description: Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-unop  |-  UniOp  =  {
t  |  ( t : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( t `  x
)  .ih  ( t `  y ) )  =  ( x  .ih  y
) ) }
Distinct variable group:    x, t, y

Detailed syntax breakdown of Definition df-unop
StepHypRef Expression
1 cuo 21545 . 2  class  UniOp
2 chil 21515 . . . . 5  class  ~H
3 vt . . . . . 6  set  t
43cv 1631 . . . . 5  class  t
52, 2, 4wfo 5269 . . . 4  wff  t : ~H -onto-> ~H
6 vx . . . . . . . . . 10  set  x
76cv 1631 . . . . . . . . 9  class  x
87, 4cfv 5271 . . . . . . . 8  class  ( t `
 x )
9 vy . . . . . . . . . 10  set  y
109cv 1631 . . . . . . . . 9  class  y
1110, 4cfv 5271 . . . . . . . 8  class  ( t `
 y )
12 csp 21518 . . . . . . . 8  class  .ih
138, 11, 12co 5874 . . . . . . 7  class  ( ( t `  x ) 
.ih  ( t `  y ) )
147, 10, 12co 5874 . . . . . . 7  class  ( x 
.ih  y )
1513, 14wceq 1632 . . . . . 6  wff  ( ( t `  x ) 
.ih  ( t `  y ) )  =  ( x  .ih  y
)
1615, 9, 2wral 2556 . . . . 5  wff  A. y  e.  ~H  ( ( t `
 x )  .ih  ( t `  y
) )  =  ( x  .ih  y )
1716, 6, 2wral 2556 . . . 4  wff  A. x  e.  ~H  A. y  e. 
~H  ( ( t `
 x )  .ih  ( t `  y
) )  =  ( x  .ih  y )
185, 17wa 358 . . 3  wff  ( t : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( t `  x
)  .ih  ( t `  y ) )  =  ( x  .ih  y
) )
1918, 3cab 2282 . 2  class  { t  |  ( t : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( t `
 x )  .ih  ( t `  y
) )  =  ( x  .ih  y ) ) }
201, 19wceq 1632 1  wff  UniOp  =  {
t  |  ( t : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( t `  x
)  .ih  ( t `  y ) )  =  ( x  .ih  y
) ) }
Colors of variables: wff set class
This definition is referenced by:  elunop  22468
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