Theorem specval 32533

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: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ})

Distinct variable group:   𝑥,𝑇

Proof of Theorem specval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 11311	. . 3 ⊢ ℂ ∈ V
2	1	rabex 5305	. 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ} ∈ V
3		ax-hilex 31634	. 2 ⊢ ℋ ∈ V
4		oveq1 7453	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ))))
5		f1eq1 6798	. . . . 5 ⊢ ((𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ))) → ((𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ ↔ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ))
6	4, 5	syl 18	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ ↔ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ))
7	6	notbid 321	. . 3 ⊢ (𝑡 = 𝑇 → (¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ ↔ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ))
8	7	rabbidv 3423	. 2 ⊢ (𝑡 = 𝑇 → {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ} = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ})
9		df-spec 32490	. 2 ⊢ Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ})
10	2, 3, 3, 8, 9	fvmptmap 8940	1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ⟶1-1 ℋ})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570  {crab 3416   I cid 5550   ↾ cres 5658  ⟶wf 6557  ⟶1-1wf1 6558  ‘cfv 6561  (class class class)co 7446  ℂcc 11228   ℋchba 31554   ·_op chot 31574   −_op chod 31575  Lambdacspc 31596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-hilex 31634

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880  df-spec 32490

This theorem is used by:  speccl  32534