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Theorem specval 26640
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 9585 . . 3  |-  CC  e.  _V
21rabex 4604 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  e.  _V
3 ax-hilex 25739 . 2  |-  ~H  e.  _V
4 oveq1 6302 . . . . 5  |-  ( t  =  T  ->  (
t  -op  ( x  .op  (  _I  |`  ~H )
) )  =  ( T  -op  ( x 
.op  (  _I  |`  ~H )
) ) )
5 f1eq1 5782 . . . . 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 3110 . 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 26597 . 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 7467 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 1379   {crab 2821    _I cid 4796    |` cres 5007   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6295   CCcc 9502   ~Hchil 25659    .op chot 25679    -op chod 25680   Lambdacspc 25701
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-hilex 25739
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-spec 26597
This theorem is referenced by:  speccl  26641
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