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Theorem specval 29282
Description: The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
specval (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
Distinct variable group:   𝑥,𝑇

Proof of Theorem specval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cnex 10305 . . 3 ℂ ∈ V
21rabex 5007 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3 ax-hilex 28381 . 2 ℋ ∈ V
4 oveq1 6885 . . . . 5 (𝑡 = 𝑇 → (𝑡op (𝑥 ·op ( I ↾ ℋ))) = (𝑇op (𝑥 ·op ( I ↾ ℋ))))
5 f1eq1 6311 . . . . 5 ((𝑡op (𝑥 ·op ( I ↾ ℋ))) = (𝑇op (𝑥 ·op ( I ↾ ℋ))) → ((𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ))
64, 5syl 17 . . . 4 (𝑡 = 𝑇 → ((𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ))
76notbid 310 . . 3 (𝑡 = 𝑇 → (¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ))
87rabbidv 3373 . 2 (𝑡 = 𝑇 → {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
9 df-spec 29239 . 2 Lambda = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
102, 3, 3, 8, 9fvmptmap 8132 1 (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198   = wceq 1653  {crab 3093   I cid 5219  cres 5314  wf 6097  1-1wf1 6098  cfv 6101  (class class class)co 6878  cc 10222  chba 28301   ·op chot 28321  op chod 28322  Lambdacspc 28343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-hilex 28381
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-spec 29239
This theorem is referenced by:  speccl  29283
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