Definition df-eigvec 30215

Description: Define the eigenvector function. Theorem eleigveccl 30321 shows that eigvec‘𝑇, the set of eigenvectors of Hilbert space operator 𝑇, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-eigvec	⊢ eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)})

Distinct variable group:   𝑥,𝑡,𝑧

Detailed syntax breakdown of Definition df-eigvec

Step	Hyp	Ref	Expression
1		cei 29321	. 2 class eigvec
2		vt	. . 3 setvar 𝑡
3		chba 29281	. . . 4 class ℋ
4		cmap 8615	. . . 4 class ↑m
5	3, 3, 4	co 7275	. . 3 class ( ℋ ↑m ℋ)
6		vx	. . . . . . . 8 setvar 𝑥
7	6	cv 1538	. . . . . . 7 class 𝑥
8	2	cv 1538	. . . . . . 7 class 𝑡
9	7, 8	cfv 6433	. . . . . 6 class (𝑡‘𝑥)
10		vz	. . . . . . . 8 setvar 𝑧
11	10	cv 1538	. . . . . . 7 class 𝑧
12		csm 29283	. . . . . . 7 class ·ℎ
13	11, 7, 12	co 7275	. . . . . 6 class (𝑧 ·ℎ 𝑥)
14	9, 13	wceq 1539	. . . . 5 wff (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)
15		cc 10869	. . . . 5 class ℂ
16	14, 10, 15	wrex 3065	. . . 4 wff ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)
17		c0h 29297	. . . . 5 class 0ℋ
18	3, 17	cdif 3884	. . . 4 class ( ℋ ∖ 0ℋ)
19	16, 6, 18	crab 3068	. . 3 class {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)}
20	2, 5, 19	cmpt 5157	. 2 class (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)})
21	1, 20	wceq 1539	1 wff eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)})

This definition is referenced by:  eigvecval  30258