Theorem eigvecval 28139

Description: The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
eigvecval	⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)})

Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem eigvecval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 27240	. . . 4 ⊢ ℋ ∈ V
2		difexg 4735	. . . 4 ⊢ ( ℋ ∈ V → ( ℋ ∖ 0ℋ) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( ℋ ∖ 0ℋ) ∈ V
4	3	rabex 4740	. 2 ⊢ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)} ∈ V
5		fveq1 6102	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
6	5	eqeq1d 2612	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
7	6	rexbidv 3034	. . 3 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
8	7	rabbidv 3164	. 2 ⊢ (𝑡 = 𝑇 → {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥)} = {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)})
9		df-eigvec 28096	. 2 ⊢ eigvec = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥)})
10	4, 1, 1, 8, 9	fvmptmap 7780	1 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)})

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   ℋchil 27160   ·_ℎ csm 27162  0_ℋc0h 27176  eigveccei 27200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-eigvec 28096

This theorem is referenced by:  eleigvec  28200