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Theorem speccl 30890
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 30889 . 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 4041 . 2 {π‘₯ ∈ β„‚ ∣ Β¬ (𝑇 βˆ’op (π‘₯ Β·op ( I β†Ύ β„‹))): ℋ–1-1β†’ β„‹} βŠ† β„‚
31, 2eqsstrdi 4002 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 3406   βŠ† wss 3914   I cid 5534   β†Ύ cres 5639  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057   β„‹chba 29910   Β·op chot 29930   βˆ’op chod 29931  Lambdacspc 29952
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-hilex 29990
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-spec 30846
This theorem is referenced by: (None)
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