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Theorem elunop 23332
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 2928 . 2  |-  ( T  e.  UniOp  ->  T  e.  _V )
2 fof 5616 . . . 4  |-  ( T : ~H -onto-> ~H  ->  T : ~H --> ~H )
3 ax-hilex 22459 . . . 4  |-  ~H  e.  _V
4 fex 5932 . . . 4  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
52, 3, 4sylancl 644 . . 3  |-  ( T : ~H -onto-> ~H  ->  T  e.  _V )
65adantr 452 . 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 5612 . . . 4  |-  ( t  =  T  ->  (
t : ~H -onto-> ~H  <->  T : ~H -onto-> ~H )
)
8 fveq1 5690 . . . . . . 7  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
9 fveq1 5690 . . . . . . 7  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
108, 9oveq12d 6062 . . . . . 6  |-  ( t  =  T  ->  (
( t `  x
)  .ih  ( t `  y ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
1110eqeq1d 2416 . . . . 5  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  (
t `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) ) )
12112ralbidv 2712 . . . 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 692 . . 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 23303 . . 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 3048 . 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 343 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   -->wf 5413   -onto->wfo 5415   ` cfv 5417  (class class class)co 6044   ~Hchil 22379    .ih csp 22382   UniOpcuo 22409
This theorem is referenced by:  unop  23375  unopf1o  23376  cnvunop  23378  counop  23381  idunop  23438  lnopunii  23472  elunop2  23473
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-hilex 22459
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-unop 23303
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