Theorem speccl 29678

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 29677	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
2		ssrab2 4058	. 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ
3	1, 2	eqsstrdi 4023	1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)

Syntax hints:  ¬ wn 3   → wi 4  {crab 3144   ⊆ wss 3938   I cid 5461   ↾ cres 5559  ⟶wf 6353  –1-1→wf1 6354  ‘cfv 6357  (class class class)co 7158  ℂcc 10537   ℋchba 28698   ·_op chot 28718   −_op chod 28719  Lambdacspc 28740

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-spec 29634

This theorem is referenced by: (None)