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Axiom ax-his1 21622
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that  * `  x is the complex conjugate cjval 11553 of  x. In the literature, the inner product of  A and  B is usually written  <. A ,  B >., but our operation notation co 5792 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 3623. Physicists use  <. B  |  A >., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 22391. (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 21460 . . . 4  class  ~H
31, 2wcel 1621 . . 3  wff  A  e. 
~H
4 cB . . . 4  class  B
54, 2wcel 1621 . . 3  wff  B  e. 
~H
63, 5wa 360 . 2  wff  ( A  e.  ~H  /\  B  e.  ~H )
7 csp 21463 . . . 4  class  .ih
81, 4, 7co 5792 . . 3  class  ( A 
.ih  B )
94, 1, 7co 5792 . . . 4  class  ( B 
.ih  A )
10 ccj 11547 . . . 4  class  *
119, 10cfv 4673 . . 3  class  ( * `
 ( B  .ih  A ) )
128, 11wceq 1619 . 2  wff  ( A 
.ih  B )  =  ( * `  ( B  .ih  A ) )
136, 12wi 6 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  21626  his7  21630  his2sub2  21633  hire  21634  hi02  21637  his1i  21640  abshicom  21641  hial2eq2  21647  orthcom  21648  adjsym  22374  cnvadj  22433  adj2  22475
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