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Axiom ax-his1 21677
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that  * `  x is the complex conjugate cjval 11603 of  x. In the literature, the inner product of  A and  B is usually written  <. A ,  B >., but our operation notation co 5874 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 3662. Physicists use  <. B  |  A >., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 22446. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-his1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( * `
 ( B  .ih  A ) ) )

Detailed syntax breakdown of Axiom ax-his1
StepHypRef Expression
1 cA . . . 4  class  A
2 chil 21515 . . . 4  class  ~H
31, 2wcel 1696 . . 3  wff  A  e. 
~H
4 cB . . . 4  class  B
54, 2wcel 1696 . . 3  wff  B  e. 
~H
63, 5wa 358 . 2  wff  ( A  e.  ~H  /\  B  e.  ~H )
7 csp 21518 . . . 4  class  .ih
81, 4, 7co 5874 . . 3  class  ( A 
.ih  B )
94, 1, 7co 5874 . . . 4  class  ( B 
.ih  A )
10 ccj 11597 . . . 4  class  *
119, 10cfv 5271 . . 3  class  ( * `
 ( B  .ih  A ) )
128, 11wceq 1632 . 2  wff  ( A 
.ih  B )  =  ( * `  ( B  .ih  A ) )
136, 12wi 4 1  wff  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( * `
 ( B  .ih  A ) ) )
Colors of variables: wff set class
This axiom is referenced by:  his5  21681  his7  21685  his2sub2  21688  hire  21689  hi02  21692  his1i  21695  abshicom  21696  hial2eq2  21702  orthcom  21703  adjsym  22429  cnvadj  22488  adj2  22530
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