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Theorem specval 29679
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 10612 . . 3 ℂ ∈ V
21rabex 5222 . 2 {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3 ax-hilex 28780 . 2 ℋ ∈ V
4 oveq1 7153 . . . . 5 (𝑡 = 𝑇 → (𝑡op (𝑥 ·op ( I ↾ ℋ))) = (𝑇op (𝑥 ·op ( I ↾ ℋ))))
5 f1eq1 6559 . . . . 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 321 . . 3 (𝑡 = 𝑇 → (¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ))
87rabbidv 3466 . 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 29636 . 2 Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
102, 3, 3, 8, 9fvmptmap 8437 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 209   = wceq 1538  {crab 3137   I cid 5447  cres 5545  wf 6340  1-1wf1 6341  cfv 6344  (class class class)co 7146  cc 10529  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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-map 8400  df-spec 29636
This theorem is referenced by:  speccl  29680
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