Theorem specval 32505

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 11284	. . 3 ⊢ ℂ ∈ V
2	1	rabex 5299	. 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3		ax-hilex 31606	. 2 ⊢ ℋ ∈ V
4		oveq1 7427	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ))))
5		f1eq1 6773	. . . . 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 3419	. 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 32462	. 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 8912	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 1569  {crab 3412   I cid 5544   ↾ cres 5652  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  ℂcc 11201   ℋchba 31526   ·_op chot 31546   −_op chod 31547  Lambdacspc 31568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-hilex 31606

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8852  df-spec 32462

This theorem is used by:  speccl  32506