Theorem specval 32231

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 11182	. . 3 ⊢ ℂ ∈ V
2	1	rabex 5311	. 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3		ax-hilex 31332	. 2 ⊢ ℋ ∈ V
4		oveq1 7419	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ))))
5		f1eq1 6771	. . . . 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 32188	. 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 8880	1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570  {crab 3416   I cid 5557   ↾ cres 5665  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7412  ℂcc 11099   ℋchba 31252   ·_op chot 31272   −_op chod 31273  Lambdacspc 31294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-hilex 31332

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-spec 32188

This theorem is referenced by:  speccl  32232