Theorem specval 30392

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 11031	. . 3 ⊢ ℂ ∈ V
2	1	rabex 5270	. 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V
3		ax-hilex 29493	. 2 ⊢ ℋ ∈ V
4		oveq1 7323	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ))))
5		f1eq1 6702	. . . . 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 17	. . . 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 317	. . 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 3411	. 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 30349	. 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 8718	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 205   = wceq 1540  {crab 3403   I cid 5505   ↾ cres 5609  ⟶wf 6461  –1-1→wf1 6462  ‘cfv 6465  (class class class)co 7316  ℂcc 10948   ℋchba 29413   ·_op chot 29433   −_op chod 29434  Lambdacspc 29455

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-hilex 29493

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-map 8666  df-spec 30349

This theorem is referenced by:  speccl  30393