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

Theorem specval 22478
Description: The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
specval  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
)
Distinct variable group:    x, T

Proof of Theorem specval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 cnex 8818 . . 3  |-  CC  e.  _V
21rabex 4165 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  e.  _V
3 ax-hilex 21579 . 2  |-  ~H  e.  _V
4 oveq1 5865 . . . . 5  |-  ( t  =  T  ->  (
t  -op  ( x  .op  (  _I  |`  ~H )
) )  =  ( T  -op  ( x 
.op  (  _I  |`  ~H )
) ) )
5 f1eq1 5432 . . . . 5  |-  ( ( t  -op  ( x 
.op  (  _I  |`  ~H )
) )  =  ( T  -op  ( x 
.op  (  _I  |`  ~H )
) )  ->  (
( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H ) )
64, 5syl 15 . . . 4  |-  ( t  =  T  ->  (
( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H ) )
76notbid 285 . . 3  |-  ( t  =  T  ->  ( -.  ( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H ) )
87rabbidv 2780 . 2  |-  ( t  =  T  ->  { x  e.  CC  |  -.  (
t  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H } )
9 df-spec 22435 . 2  |-  Lambda  =  ( t  e.  ( ~H 
^m  ~H )  |->  { x  e.  CC  |  -.  (
t  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H } )
102, 3, 3, 8, 9fvmptmap 6804 1  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623   {crab 2547    _I cid 4304    |` cres 4691   -->wf 5251   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    .op chot 21519    -op chod 21520   Lambdacspc 21541
This theorem is referenced by:  speccl  22479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-spec 22435
  Copyright terms: Public domain W3C validator