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Theorem specval 31138
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 11187 . . 3 β„‚ ∈ V
21rabex 5331 . 2 {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} ∈ V
3 ax-hilex 30239 . 2 β„‹ ∈ V
4 oveq1 7412 . . . . 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 6779 . . . . 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 3440 . 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 31095 . 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 8871 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 1541  {crab 3432   I cid 5572   β†Ύ cres 5677  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104   β„‹chba 30159   Β·op chot 30179   βˆ’op chod 30180  Lambdacspc 30201
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-hilex 30239
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-spec 31095
This theorem is referenced by:  speccl  31139
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