Theorem eleigvec 31645

Description: Membership in 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
eleigvec	⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem eleigvec

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eigvecval 31584	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)})
2	1	eleq2d 2811	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)}))
3		eldif 3950	. . . . 5 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ))
4		elch0 30942	. . . . . . 7 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)
5	4	necon3bbii 2980	. . . . . 6 ⊢ (¬ 𝐴 ∈ 0ℋ ↔ 𝐴 ≠ 0ℎ)
6	5	anbi2i 622	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ))
7	3, 6	bitri 275	. . . 4 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ))
8	7	anbi1i 623	. . 3 ⊢ ((𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
9		fveq2 6881	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘𝑦) = (𝑇‘𝐴))
10		oveq2 7409	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 𝐴))
11	9, 10	eqeq12d 2740	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
12	11	rexbidv 3170	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
13	12	elrab 3675	. . 3 ⊢ (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
14		df-3an 1086	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
15	8, 13, 14	3bitr4i 303	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)} ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
16	2, 15	bitrdi 287	1 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∃wrex 3062  {crab 3424   ∖ cdif 3937  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  ℂcc 11103   ℋchba 30607   ·_ℎ csm 30609  0_ℎc0v 30612  0_ℋc0h 30623  eigveccei 30647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-hilex 30687  ax-hv0cl 30691

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-ch0 30941  df-eigvec 31541

This theorem is referenced by:  eleigvec2  31646  eigvalcl  31649