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Theorem speccl 31918
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 31917 . 2 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
2 ssrab2 4080 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ
31, 2eqsstrdi 4028 1 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  {crab 3436  wss 3951   I cid 5577  cres 5687  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  cc 11153  chba 30938   ·op chot 30958  op chod 30959  Lambdacspc 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-spec 31874
This theorem is referenced by: (None)
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