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

Theorem specval 22533
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 8863 . . 3  |-  CC  e.  _V
21rabex 4202 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  e.  _V
3 ax-hilex 21634 . 2  |-  ~H  e.  _V
4 oveq1 5907 . . . . 5  |-  ( t  =  T  ->  (
t  -op  ( x  .op  (  _I  |`  ~H )
) )  =  ( T  -op  ( x 
.op  (  _I  |`  ~H )
) ) )
5 f1eq1 5470 . . . . 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 2814 . 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 22490 . 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 6847 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 1633   {crab 2581    _I cid 4341    |` cres 4728   -->wf 5288   -1-1->wf1 5289   ` cfv 5292  (class class class)co 5900   CCcc 8780   ~Hchil 21554    .op chot 21574    -op chod 21575   Lambdacspc 21596
This theorem is referenced by:  speccl  22534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-hilex 21634
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-spec 22490
  Copyright terms: Public domain W3C validator