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Definition df-kb 22447
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 22446, 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 21553 . 2  class  ketbra
2 vx . . 3  set  x
3 vy . . 3  set  y
4 chil 21515 . . 3  class  ~H
5 vz . . . 4  set  z
65cv 1631 . . . . . 6  class  z
73cv 1631 . . . . . 6  class  y
8 csp 21518 . . . . . 6  class  .ih
96, 7, 8co 5874 . . . . 5  class  ( z 
.ih  y )
102cv 1631 . . . . 5  class  x
11 csm 21517 . . . . 5  class  .h
129, 10, 11co 5874 . . . 4  class  ( ( z  .ih  y )  .h  x )
135, 4, 12cmpt 4093 . . 3  class  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) )
142, 3, 4, 4, 13cmpt2 5876 . 2  class  ( x  e.  ~H ,  y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) ) )
151, 14wceq 1632 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  22548
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