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Theorem specval 31655
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 11190 . . 3 β„‚ ∈ V
21rabex 5325 . 2 {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} ∈ V
3 ax-hilex 30756 . 2 β„‹ ∈ V
4 oveq1 7411 . . . . 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 6775 . . . . 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 318 . . 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 3434 . 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 31612 . 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 8874 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 3426   I cid 5566   β†Ύ cres 5671  βŸΆwf 6532  β€“1-1β†’wf1 6533  β€˜cfv 6536  (class class class)co 7404  β„‚cc 11107   β„‹chba 30676   Β·op chot 30696   βˆ’op chod 30697  Lambdacspc 30718
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-hilex 30756
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-spec 31612
This theorem is referenced by:  speccl  31656
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