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Theorem elhmop 22453
Description: Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elhmop  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
Distinct variable group:    x, y, T

Proof of Theorem elhmop
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq1 5524 . . . . . 6  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
21oveq2d 5874 . . . . 5  |-  ( t  =  T  ->  (
x  .ih  ( t `  y ) )  =  ( x  .ih  ( T `  y )
) )
3 fveq1 5524 . . . . . 6  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
43oveq1d 5873 . . . . 5  |-  ( t  =  T  ->  (
( t `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
52, 4eqeq12d 2297 . . . 4  |-  ( t  =  T  ->  (
( x  .ih  (
t `  y )
)  =  ( ( t `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
) )
652ralbidv 2585 . . 3  |-  ( t  =  T  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( t `  y ) )  =  ( ( t `  x )  .ih  y
)  <->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) ) )
7 df-hmop 22424 . . 3  |-  HrmOp  =  {
t  e.  ( ~H 
^m  ~H )  |  A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( t `  y
) )  =  ( ( t `  x
)  .ih  y ) }
86, 7elrab2 2925 . 2  |-  ( T  e.  HrmOp 
<->  ( T  e.  ( ~H  ^m  ~H )  /\  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) ) )
9 ax-hilex 21579 . . . 4  |-  ~H  e.  _V
109, 9elmap 6796 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1110anbi1i 676 . 2  |-  ( ( T  e.  ( ~H 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( T `  y ) )  =  ( ( T `  x ) 
.ih  y ) )  <-> 
( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
128, 11bitri 240 1  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   ~Hchil 21499    .ih csp 21502   HrmOpcho 21530
This theorem is referenced by:  hmopf  22454  hmop  22502  hmopadj2  22521  idhmop  22562  0hmop  22563  lnophmi  22598  hmops  22600  hmopm  22601  hmopco  22603  pjhmopi  22726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hmop 22424
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