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Theorem specval 29062
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 10205 . . 3 ℂ ∈ V
21rabex 4960 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3 ax-hilex 28161 . 2 ℋ ∈ V
4 oveq1 6816 . . . . 5 (𝑡 = 𝑇 → (𝑡op (𝑥 ·op ( I ↾ ℋ))) = (𝑇op (𝑥 ·op ( I ↾ ℋ))))
5 f1eq1 6253 . . . . 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 307 . . 3 (𝑡 = 𝑇 → (¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ))
87rabbidv 3325 . 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 29019 . 2 Lambda = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
102, 3, 3, 8, 9fvmptmap 8056 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 196   = wceq 1628  {crab 3050   I cid 5169  cres 5264  wf 6041  1-1wf1 6042  cfv 6045  (class class class)co 6809  cc 10122  chil 28081   ·op chot 28101  op chod 28102  Lambdacspc 28123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-hilex 28161
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-map 8021  df-spec 29019
This theorem is referenced by:  speccl  29063
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