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Theorem elunop 27401
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 3087 . 2  |-  ( T  e.  UniOp  ->  T  e.  _V )
2 fof 5801 . . . 4  |-  ( T : ~H -onto-> ~H  ->  T : ~H --> ~H )
3 ax-hilex 26528 . . . 4  |-  ~H  e.  _V
4 fex 6144 . . . 4  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
52, 3, 4sylancl 666 . . 3  |-  ( T : ~H -onto-> ~H  ->  T  e.  _V )
65adantr 466 . 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 5797 . . . 4  |-  ( t  =  T  ->  (
t : ~H -onto-> ~H  <->  T : ~H -onto-> ~H )
)
8 fveq1 5871 . . . . . . 7  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
9 fveq1 5871 . . . . . . 7  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
108, 9oveq12d 6314 . . . . . 6  |-  ( t  =  T  ->  (
( t `  x
)  .ih  ( t `  y ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
1110eqeq1d 2422 . . . . 5  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  (
t `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) ) )
12112ralbidv 2867 . . . 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 715 . . 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 27372 . . 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 3217 . 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 354 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 setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078   -->wf 5588   -onto->wfo 5590   ` cfv 5592  (class class class)co 6296   ~Hchil 26448    .ih csp 26451   UniOpcuo 26478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-hilex 26528
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-unop 27372
This theorem is referenced by:  unop  27444  unopf1o  27445  cnvunop  27447  counop  27450  idunop  27507  lnopunii  27541  elunop2  27542
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