Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  eigvecval Structured version   Visualization version   GIF version

Theorem eigvecval 29683
 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 ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem eigvecval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28786 . . . 4 ℋ ∈ V
2 difexg 5198 . . . 4 ( ℋ ∈ V → ( ℋ ∖ 0) ∈ V)
31, 2ax-mp 5 . . 3 ( ℋ ∖ 0) ∈ V
43rabex 5202 . 2 {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)} ∈ V
5 fveq1 6648 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65eqeq1d 2803 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) = (𝑦 · 𝑥) ↔ (𝑇𝑥) = (𝑦 · 𝑥)))
76rexbidv 3259 . . 3 (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)))
87rabbidv 3430 . 2 (𝑡 = 𝑇 → {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)} = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
9 df-eigvec 29640 . 2 eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)})
104, 1, 1, 8, 9fvmptmap 8432 1 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  ∃wrex 3110  {crab 3113  Vcvv 3444   ∖ cdif 3881  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  ℂcc 10528   ℋchba 28706   ·ℎ csm 28708  0ℋc0h 28722  eigveccei 28746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-hilex 28786 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-eigvec 29640 This theorem is referenced by:  eleigvec  29744
 Copyright terms: Public domain W3C validator