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Theorem speccl 31927
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)

Proof of Theorem speccl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 specval 31926 . 2 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
2 ssrab2 4089 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ
31, 2eqsstrdi 4049 1 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  {crab 3432  wss 3962   I cid 5581  cres 5690  wf 6558  1-1wf1 6559  cfv 6562  (class class class)co 7430  cc 11150  chba 30947   ·op chot 30967  op chod 30968  Lambdacspc 30989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-hilex 31027
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-spec 31883
This theorem is referenced by: (None)
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