Theorem eleigvec 31892

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 31831	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)})
2	1	eleq2d 2815	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)}))
3		eldif 3926	. . . . 5 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ))
4		elch0 31189	. . . . . . 7 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)
5	4	necon3bbii 2973	. . . . . 6 ⊢ (¬ 𝐴 ∈ 0ℋ ↔ 𝐴 ≠ 0ℎ)
6	5	anbi2i 623	. . . . 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 624	. . 3 ⊢ ((𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
9		fveq2 6860	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘𝑦) = (𝑇‘𝐴))
10		oveq2 7397	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 𝐴))
11	9, 10	eqeq12d 2746	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
12	11	rexbidv 3158	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))
13	12	elrab 3661	. . 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 1088	. . 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  {crab 3408   ∖ cdif 3913  ⟶wf 6509  ‘cfv 6513  (class class class)co 7389  ℂcc 11072   ℋchba 30854   ·_ℎ csm 30856  0_ℎc0v 30859  0_ℋc0h 30870  eigveccei 30894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-hilex 30934  ax-hv0cl 30938

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-map 8803  df-ch0 31188  df-eigvec 31788

This theorem is referenced by:  eleigvec2  31893  eigvalcl  31896