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Theorem elunop 23380
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 2966 . 2  |-  ( T  e.  UniOp  ->  T  e.  _V )
2 fof 5656 . . . 4  |-  ( T : ~H -onto-> ~H  ->  T : ~H --> ~H )
3 ax-hilex 22507 . . . 4  |-  ~H  e.  _V
4 fex 5972 . . . 4  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
52, 3, 4sylancl 645 . . 3  |-  ( T : ~H -onto-> ~H  ->  T  e.  _V )
65adantr 453 . 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 5652 . . . 4  |-  ( t  =  T  ->  (
t : ~H -onto-> ~H  <->  T : ~H -onto-> ~H )
)
8 fveq1 5730 . . . . . . 7  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
9 fveq1 5730 . . . . . . 7  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
108, 9oveq12d 6102 . . . . . 6  |-  ( t  =  T  ->  (
( t `  x
)  .ih  ( t `  y ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
1110eqeq1d 2446 . . . . 5  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  (
t `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) ) )
12112ralbidv 2749 . . . 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 693 . . 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 23351 . . 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 3086 . 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 344 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   -->wf 5453   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084   ~Hchil 22427    .ih csp 22430   UniOpcuo 22457
This theorem is referenced by:  unop  23423  unopf1o  23424  cnvunop  23426  counop  23429  idunop  23486  lnopunii  23520  elunop2  23521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-hilex 22507
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-unop 23351
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