Theorem speccl 30261

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: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)

Proof of Theorem speccl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		specval 30260	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
2		ssrab2 4013	. 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ
3	1, 2	eqsstrdi 3975	1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)

Syntax hints:  ¬ wn 3   → wi 4  {crab 3068   ⊆ wss 3887   I cid 5488   ↾ cres 5591  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  (class class class)co 7275  ℂcc 10869   ℋchba 29281   ·_op chot 29301   −_op chod 29302  Lambdacspc 29323

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-hilex 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-spec 30217

This theorem is referenced by: (None)