HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ax-his1 Structured version   Unicode version

Axiom ax-his1 26197
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that  * `  x is the complex conjugate cjval 13017 of  x. In the literature, the inner product of  A and  B is usually written  <. A ,  B >., but our operation notation co 6270 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4023. Physicists use  <. B  |  A >., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 26967. (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 26034 . . . 4  class  ~H
31, 2wcel 1823 . . 3  wff  A  e. 
~H
4 cB . . . 4  class  B
54, 2wcel 1823 . . 3  wff  B  e. 
~H
63, 5wa 367 . 2  wff  ( A  e.  ~H  /\  B  e.  ~H )
7 csp 26037 . . . 4  class  .ih
81, 4, 7co 6270 . . 3  class  ( A 
.ih  B )
94, 1, 7co 6270 . . . 4  class  ( B 
.ih  A )
10 ccj 13011 . . . 4  class  *
119, 10cfv 5570 . . 3  class  ( * `
 ( B  .ih  A ) )
128, 11wceq 1398 . 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 setvar class
This axiom is referenced by:  his5  26201  his7  26205  his2sub2  26208  hire  26209  hi02  26212  his1i  26215  abshicom  26216  hial2eq2  26222  orthcom  26223  adjsym  26950  cnvadj  27009  adj2  27051
  Copyright terms: Public domain W3C validator