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Theorem speccl 30945
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 30944 . 2 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
2 ssrab2 4064 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ
31, 2eqsstrdi 4023 1 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  {crab 3425  wss 3935   I cid 5557  cres 5662  wf 6519  1-1wf1 6520  cfv 6523  (class class class)co 7384  cc 11080  chba 29965   ·op chot 29985  op chod 29986  Lambdacspc 30007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pow 5347  ax-pr 5411  ax-un 7699  ax-cnex 11138  ax-hilex 30045
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3426  df-v 3468  df-sbc 3765  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-br 5133  df-opab 5195  df-mpt 5216  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fv 6531  df-ov 7387  df-oprab 7388  df-mpo 7389  df-map 8796  df-spec 30901
This theorem is referenced by: (None)
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