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Theorem speccl 29680
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 29679 . 2 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
2 ssrab2 4031 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ
31, 2eqsstrdi 3996 1 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  {crab 3134  wss 3908   I cid 5436  cres 5534  wf 6330  1-1wf1 6331  cfv 6334  (class class class)co 7140  cc 10524  chba 28700   ·op chot 28720  op chod 28721  Lambdacspc 28742
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-hilex 28780
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-spec 29636
This theorem is referenced by: (None)
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