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

Theorem idunop 23473
Description: The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
idunop  |-  (  _I  |`  ~H )  e.  UniOp

Proof of Theorem idunop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5705 . . 3  |-  (  _I  |`  ~H ) : ~H -1-1-onto-> ~H
2 f1ofo 5673 . . 3  |-  ( (  _I  |`  ~H ) : ~H -1-1-onto-> ~H  ->  (  _I  |` 
~H ) : ~H -onto-> ~H )
31, 2ax-mp 8 . 2  |-  (  _I  |`  ~H ) : ~H -onto-> ~H
4 fvresi 5916 . . . 4  |-  ( x  e.  ~H  ->  (
(  _I  |`  ~H ) `  x )  =  x )
5 fvresi 5916 . . . 4  |-  ( y  e.  ~H  ->  (
(  _I  |`  ~H ) `  y )  =  y )
64, 5oveqan12d 6092 . . 3  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( (  _I  |`  ~H ) `  x
)  .ih  ( (  _I  |`  ~H ) `  y ) )  =  ( x  .ih  y
) )
76rgen2a 2764 . 2  |-  A. x  e.  ~H  A. y  e. 
~H  ( ( (  _I  |`  ~H ) `  x )  .ih  (
(  _I  |`  ~H ) `  y ) )  =  ( x  .ih  y
)
8 elunop 23367 . 2  |-  ( (  _I  |`  ~H )  e.  UniOp 
<->  ( (  _I  |`  ~H ) : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( (  _I  |`  ~H ) `  x )  .ih  (
(  _I  |`  ~H ) `  y ) )  =  ( x  .ih  y
) ) )
93, 7, 8mpbir2an 887 1  |-  (  _I  |`  ~H )  e.  UniOp
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   A.wral 2697    _I cid 4485    |` cres 4872   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   ~Hchil 22414    .ih csp 22417   UniOpcuo 22444
This theorem is referenced by:  idlnop  23487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-unop 23338
  Copyright terms: Public domain W3C validator