HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  specval Structured version   Visualization version   GIF version

Theorem specval 28885
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 10055 . . 3 ℂ ∈ V
21rabex 4845 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3 ax-hilex 27984 . 2 ℋ ∈ V
4 oveq1 6697 . . . . 5 (𝑡 = 𝑇 → (𝑡op (𝑥 ·op ( I ↾ ℋ))) = (𝑇op (𝑥 ·op ( I ↾ ℋ))))
5 f1eq1 6134 . . . . 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 3220 . 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 28842 . 2 Lambda = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
102, 3, 3, 8, 9fvmptmap 7936 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 1523  {crab 2945   I cid 5052  cres 5145  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  cc 9972  chil 27904   ·op chot 27924  op chod 27925  Lambdacspc 27946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-spec 28842
This theorem is referenced by:  speccl  28886
  Copyright terms: Public domain W3C validator