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Theorem specval 31728
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: β„‹βŸΆ β„‹ β†’ (Lambdaβ€˜π‘‡) = {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹})
Distinct variable group:   π‘₯,𝑇
Proof of Theorem specval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 cnex 11227 . . 3 β„‚ ∈ V
21rabex 5338 . 2 {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} ∈ V
3 ax-hilex 30829 . 2 β„‹ ∈ V
4 oveq1 7433 . . . . 5 (𝑑 = 𝑇 β†’ (𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))) = (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))))
5 f1eq1 6793 . . . . 5 ((𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))) = (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))) β†’ ((𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹ ↔ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹))
64, 5syl 17 . . . 4 (𝑑 = 𝑇 β†’ ((𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹ ↔ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹))
76notbid 317 . . 3 (𝑑 = 𝑇 β†’ (Β¬ (𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹ ↔ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹))
87rabbidv 3438 . 2 (𝑑 = 𝑇 β†’ {π‘₯ ∈ β„‚ ∣ Β¬ (𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} = {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹})
9 df-spec 31685 . 2 Lambda = (𝑑 ∈ ( β„‹ ↑m β„‹) ↦ {π‘₯ ∈ β„‚ ∣ Β¬ (𝑑 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹})
102, 3, 3, 8, 9fvmptmap 8906 1 (𝑇: β„‹βŸΆ β„‹ β†’ (Lambdaβ€˜π‘‡) = {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   = wceq 1533  {crab 3430   I cid 5579   β†Ύ cres 5684  βŸΆwf 6549  β€“1-1β†’wf1 6550  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11144   β„‹chba 30749   Β·op chot 30769   βˆ’op chod 30770  Lambdacspc 30791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-hilex 30829
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-spec 31685
This theorem is referenced by:  speccl  31729
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