Theorem eigvecval 31992

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 31095	. . . 4 ⊢ ℋ ∈ V
2		difexg 5264	. . . 4 ⊢ ( ℋ ∈ V → ( ℋ ∖ 0ℋ) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( ℋ ∖ 0ℋ) ∈ V
4	3	rabex 5274	. 2 ⊢ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)} ∈ V
5		fveq1 6833	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
6	5	eqeq1d 2742	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
7	6	rexbidv 3164	. . 3 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
8	7	rabbidv 3399	. 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 31949	. 2 ⊢ eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥)})
10	4, 1, 1, 8, 9	fvmptmap 8826	1 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)})

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  {crab 3392  Vcvv 3432   ∖ cdif 3887  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  ℂcc 11034   ℋchba 31015   ·_ℎ csm 31017  0_ℋc0h 31031  eigveccei 31055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-hilex 31095

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-eigvec 31949

This theorem is referenced by:  eleigvec  32053