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Theorem specval 30882
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 11139 . . 3 β„‚ ∈ V
21rabex 5294 . 2 {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} ∈ V
3 ax-hilex 29983 . 2 β„‹ ∈ V
4 oveq1 7369 . . . . 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 6738 . . . . 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 3418 . 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 30839 . 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 8826 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 1542  {crab 3410   I cid 5535   β†Ύ cres 5640  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056   β„‹chba 29903   Β·op chot 29923   βˆ’op chod 29924  Lambdacspc 29945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-hilex 29983
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-spec 30839
This theorem is referenced by:  speccl  30883
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