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Theorem elhmop 23217
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 5660 . . . . . 6  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
21oveq2d 6029 . . . . 5  |-  ( t  =  T  ->  (
x  .ih  ( t `  y ) )  =  ( x  .ih  ( T `  y )
) )
3 fveq1 5660 . . . . . 6  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
43oveq1d 6028 . . . . 5  |-  ( t  =  T  ->  (
( t `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
52, 4eqeq12d 2394 . . . 4  |-  ( t  =  T  ->  (
( x  .ih  (
t `  y )
)  =  ( ( t `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
) )
652ralbidv 2684 . . 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 23188 . . 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 3030 . 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 22343 . . . 4  |-  ~H  e.  _V
109, 9elmap 6971 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1110anbi1i 677 . 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 241 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   -->wf 5383   ` cfv 5387  (class class class)co 6013    ^m cmap 6947   ~Hchil 22263    .ih csp 22266   HrmOpcho 22294
This theorem is referenced by:  hmopf  23218  hmop  23266  hmopadj2  23285  idhmop  23326  0hmop  23327  lnophmi  23362  hmops  23364  hmopm  23365  hmopco  23367  pjhmopi  23490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-hmop 23188
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