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Definition df-kb 22377
Description: Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation,  |  A >.  <. B  | is an operator known as the outer product of  A and  B, which we represent by  ( A  ketbra  B ). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with definition df-bra 22376, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-kb  |-  ketbra  =  ( x  e.  ~H , 
y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y
)  .h  x ) ) )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-kb
StepHypRef Expression
1 ck 21483 . 2  class  ketbra
2 vx . . 3  set  x
3 vy . . 3  set  y
4 chil 21445 . . 3  class  ~H
5 vz . . . 4  set  z
65cv 1618 . . . . . 6  class  z
73cv 1618 . . . . . 6  class  y
8 csp 21448 . . . . . 6  class  .ih
96, 7, 8co 5778 . . . . 5  class  ( z 
.ih  y )
102cv 1618 . . . . 5  class  x
11 csm 21447 . . . . 5  class  .h
129, 10, 11co 5778 . . . 4  class  ( ( z  .ih  y )  .h  x )
135, 4, 12cmpt 4037 . . 3  class  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) )
142, 3, 4, 4, 13cmpt2 5780 . 2  class  ( x  e.  ~H ,  y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) ) )
151, 14wceq 1619 1  wff  ketbra  =  ( x  e.  ~H , 
y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y
)  .h  x ) ) )
Colors of variables: wff set class
This definition is referenced by:  kbfval  22478
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