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Theorem speccl 23252
Description: The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
speccl  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  C_  CC )

Proof of Theorem speccl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 specval 23251 . 2  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
)
2 ssrab2 3373 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  C_  CC
31, 2syl6eqss 3343 1  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  C_  CC )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   {crab 2655    C_ wss 3265    _I cid 4436    |` cres 4822   -->wf 5392   -1-1->wf1 5393   ` cfv 5396  (class class class)co 6022   CCcc 8923   ~Hchil 22272    .op chot 22292    -op chod 22293   Lambdacspc 22314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-hilex 22352
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-spec 23208
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