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Theorem specval 25481
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 9478 . . 3  |-  CC  e.  _V
21rabex 4554 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  e.  _V
3 ax-hilex 24580 . 2  |-  ~H  e.  _V
4 oveq1 6210 . . . . 5  |-  ( t  =  T  ->  (
t  -op  ( x  .op  (  _I  |`  ~H )
) )  =  ( T  -op  ( x 
.op  (  _I  |`  ~H )
) ) )
5 f1eq1 5712 . . . . 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 16 . . . 4  |-  ( t  =  T  ->  (
( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H ) )
76notbid 294 . . 3  |-  ( t  =  T  ->  ( -.  ( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H ) )
87rabbidv 3070 . 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 25438 . 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 7362 1  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1370   {crab 2803    _I cid 4742    |` cres 4953   -->wf 5525   -1-1->wf1 5526   ` cfv 5529  (class class class)co 6203   CCcc 9395   ~Hchil 24500    .op chot 24520    -op chod 24521   Lambdacspc 24542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-hilex 24580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-spec 25438
This theorem is referenced by:  speccl  25482
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