HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  elunop Structured version   Unicode version

Theorem elunop 26564
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 3122 . 2  |-  ( T  e.  UniOp  ->  T  e.  _V )
2 fof 5795 . . . 4  |-  ( T : ~H -onto-> ~H  ->  T : ~H --> ~H )
3 ax-hilex 25689 . . . 4  |-  ~H  e.  _V
4 fex 6134 . . . 4  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
52, 3, 4sylancl 662 . . 3  |-  ( T : ~H -onto-> ~H  ->  T  e.  _V )
65adantr 465 . 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 5791 . . . 4  |-  ( t  =  T  ->  (
t : ~H -onto-> ~H  <->  T : ~H -onto-> ~H )
)
8 fveq1 5865 . . . . . . 7  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
9 fveq1 5865 . . . . . . 7  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
108, 9oveq12d 6303 . . . . . 6  |-  ( t  =  T  ->  (
( t `  x
)  .ih  ( t `  y ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
1110eqeq1d 2469 . . . . 5  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  (
t `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) ) )
12112ralbidv 2908 . . . 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 710 . . 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 26535 . . 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 3252 . 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 353 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6285   ~Hchil 25609    .ih csp 25612   UniOpcuo 25639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-unop 26535
This theorem is referenced by:  unop  26607  unopf1o  26608  cnvunop  26610  counop  26613  idunop  26670  lnopunii  26704  elunop2  26705
  Copyright terms: Public domain W3C validator