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Theorem elunop 22560
Description: Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
elunop  |-  ( T  e.  UniOp 
<->  ( T : ~H -onto-> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y ) ) )
Distinct variable group:    x, y, T

Proof of Theorem elunop
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 elex 2872 . 2  |-  ( T  e.  UniOp  ->  T  e.  _V )
2 fof 5531 . . . 4  |-  ( T : ~H -onto-> ~H  ->  T : ~H --> ~H )
3 ax-hilex 21687 . . . 4  |-  ~H  e.  _V
4 fex 5832 . . . 4  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
52, 3, 4sylancl 643 . . 3  |-  ( T : ~H -onto-> ~H  ->  T  e.  _V )
65adantr 451 . 2  |-  ( ( T : ~H -onto-> ~H  /\ 
A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  ( T `  y ) )  =  ( x  .ih  y
) )  ->  T  e.  _V )
7 foeq1 5527 . . . 4  |-  ( t  =  T  ->  (
t : ~H -onto-> ~H  <->  T : ~H -onto-> ~H )
)
8 fveq1 5604 . . . . . . 7  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
9 fveq1 5604 . . . . . . 7  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
108, 9oveq12d 5960 . . . . . 6  |-  ( t  =  T  ->  (
( t `  x
)  .ih  ( t `  y ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
1110eqeq1d 2366 . . . . 5  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  (
t `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) ) )
12112ralbidv 2661 . . . 4  |-  ( t  =  T  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
( t `  x
)  .ih  ( t `  y ) )  =  ( x  .ih  y
)  <->  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  ( T `  y ) )  =  ( x  .ih  y
) ) )
137, 12anbi12d 691 . . 3  |-  ( t  =  T  ->  (
( t : ~H -onto-> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( t `  x )  .ih  (
t `  y )
)  =  ( x 
.ih  y ) )  <-> 
( T : ~H -onto-> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y ) ) ) )
14 df-unop 22531 . . 3  |-  UniOp  =  {
t  |  ( t : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( t `  x
)  .ih  ( t `  y ) )  =  ( x  .ih  y
) ) }
1513, 14elab2g 2992 . 2  |-  ( T  e.  _V  ->  ( T  e.  UniOp  <->  ( T : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  ( T `  y ) )  =  ( x 
.ih  y ) ) ) )
161, 6, 15pm5.21nii 342 1  |-  ( T  e.  UniOp 
<->  ( T : ~H -onto-> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   -->wf 5330   -onto->wfo 5332   ` cfv 5334  (class class class)co 5942   ~Hchil 21607    .ih csp 21610   UniOpcuo 21637
This theorem is referenced by:  unop  22603  unopf1o  22604  cnvunop  22606  counop  22609  idunop  22666  lnopunii  22700  elunop2  22701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-hilex 21687
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-unop 22531
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