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Theorem speccl 31586
Description: The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
speccl (𝑇: β„‹βŸΆ β„‹ β†’ (Lambdaβ€˜π‘‡) βŠ† β„‚)

Proof of Theorem speccl
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 specval 31585 . 2 (𝑇: β„‹βŸΆ β„‹ β†’ (Lambdaβ€˜π‘‡) = {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹})
2 ssrab2 4077 . 2 {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} βŠ† β„‚
31, 2eqsstrdi 4036 1 (𝑇: β„‹βŸΆ β„‹ β†’ (Lambdaβ€˜π‘‡) βŠ† β„‚)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4  {crab 3431   βŠ† wss 3948   I cid 5573   β†Ύ cres 5678  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11114   β„‹chba 30606   Β·op chot 30626   βˆ’op chod 30627  Lambdacspc 30648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-hilex 30686
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-spec 31542
This theorem is referenced by: (None)
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