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Theorem specval 27944
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 9870 . . 3 ℂ ∈ V
21rabex 4732 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3 ax-hilex 27043 . 2 ℋ ∈ V
4 oveq1 6531 . . . . 5 (𝑡 = 𝑇 → (𝑡op (𝑥 ·op ( I ↾ ℋ))) = (𝑇op (𝑥 ·op ( I ↾ ℋ))))
5 f1eq1 5991 . . . . 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 306 . . 3 (𝑡 = 𝑇 → (¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ))
87rabbidv 3160 . 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 27901 . 2 Lambda = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
102, 3, 3, 8, 9fvmptmap 7754 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 194   = wceq 1474  {crab 2896   I cid 4935  cres 5027  wf 5783  1-1wf1 5784  cfv 5787  (class class class)co 6524  cc 9787  chil 26963   ·op chot 26983  op chod 26984  Lambdacspc 27005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-hilex 27043
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-map 7720  df-spec 27901
This theorem is referenced by:  speccl  27945
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