Theorem eigvecval 28755

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 27856	. . . 4 ⊢ ℋ ∈ V
2		difexg 4808	. . . 4 ⊢ ( ℋ ∈ V → ( ℋ ∖ 0ℋ) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( ℋ ∖ 0ℋ) ∈ V
4	3	rabex 4813	. 2 ⊢ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)} ∈ V
5		fveq1 6190	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
6	5	eqeq1d 2624	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
7	6	rexbidv 3052	. . 3 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
8	7	rabbidv 3189	. 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 28712	. 2 ⊢ eigvec = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥)})
10	4, 1, 1, 8, 9	fvmptmap 7894	1 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)})

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℂcc 9934   ℋchil 27776   ·_ℎ csm 27778  0_ℋc0h 27792  eigveccei 27816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-eigvec 28712

This theorem is referenced by:  eleigvec  28816