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Theorem specval 27536
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 9620 . . 3  |-  CC  e.  _V
21rabex 4571 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  e.  _V
3 ax-hilex 26637 . 2  |-  ~H  e.  _V
4 oveq1 6308 . . . . 5  |-  ( t  =  T  ->  (
t  -op  ( x  .op  (  _I  |`  ~H )
) )  =  ( T  -op  ( x 
.op  (  _I  |`  ~H )
) ) )
5 f1eq1 5787 . . . . 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 17 . . . 4  |-  ( t  =  T  ->  (
( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H ) )
76notbid 295 . . 3  |-  ( t  =  T  ->  ( -.  ( t  -op  (
x  .op  (  _I  |` 
~H ) ) ) : ~H -1-1-> ~H  <->  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H ) )
87rabbidv 3072 . 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 27493 . 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 7512 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 187    = wceq 1437   {crab 2779    _I cid 4759    |` cres 4851   -->wf 5593   -1-1->wf1 5594   ` cfv 5597  (class class class)co 6301   CCcc 9537   ~Hchil 26557    .op chot 26577    -op chod 26578   Lambdacspc 26599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-spec 27493
This theorem is referenced by:  speccl  27537
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