Theorem eigvecval 31871

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 30974	. . . 4 ⊢ ℋ ∈ V
2		difexg 5267	. . . 4 ⊢ ( ℋ ∈ V → ( ℋ ∖ 0ℋ) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( ℋ ∖ 0ℋ) ∈ V
4	3	rabex 5277	. 2 ⊢ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)} ∈ V
5		fveq1 6821	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
6	5	eqeq1d 2733	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
7	6	rexbidv 3156	. . 3 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)))
8	7	rabbidv 3402	. 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 31828	. 2 ⊢ eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·ℎ 𝑥)})
10	4, 1, 1, 8, 9	fvmptmap 8805	1 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  Vcvv 3436   ∖ cdif 3899  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℂcc 11001   ℋchba 30894   ·_ℎ csm 30896  0_ℋc0h 30910  eigveccei 30934

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-hilex 30974

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-eigvec 31828

This theorem is referenced by:  eleigvec  31932