Definition df-eigvec 9779

Description: Define the eigenvector function. Theorem eleigvecclt 9883 shows that eigvec` T, the set of eigenvectors of Hilbert space operator T, are Hilbert space vectors.

Assertion

Ref	Expression
df-eigvec	|- eigvec = {<.t, y>. | (t:H~-->H~ /\ y = {x e. H~ | (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))})}

Distinct variable group:   x,t,y,z

Detailed syntax breakdown of Definition df-eigvec

Step	Hyp	Ref	Expression
1		cei 8828	. 2 class eigvec
2		chil 8788	. . . . 5 class H~
3		vt	. . . . . 6 set t
4	3	cv 955	. . . . 5 class t
5	2, 2, 4	wf 3178	. . . 4 wff t:H~-->H~
6		vy	. . . . . 6 set y
7	6	cv 955	. . . . 5 class y
8		vx	. . . . . . . . 9 set x
9	8	cv 955	. . . . . . . 8 class x
10		c0v 8791	. . . . . . . 8 class 0h
11	9, 10	wne 1585	. . . . . . 7 wff x =/= 0h
12	9, 4	cfv 3182	. . . . . . . . 9 class (t` x)
13		vz	. . . . . . . . . . 11 set z
14	13	cv 955	. . . . . . . . . 10 class z
15		csm 8790	. . . . . . . . . 10 class .h
16	14, 9, 15	co 3963	. . . . . . . . 9 class (z .h x)
17	12, 16	wceq 956	. . . . . . . 8 wff (t` x) = (z .h x)
18		cc 5232	. . . . . . . 8 class CC
19	17, 13, 18	wrex 1646	. . . . . . 7 wff E.z e. CC (t` x) = (z .h x)
20	11, 19	wa 223	. . . . . 6 wff (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))
21	20, 8, 2	crab 1648	. . . . 5 class {x e. H~ | (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))}
22	7, 21	wceq 956	. . . 4 wff y = {x e. H~ | (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))}
23	5, 22	wa 223	. . 3 wff (t:H~-->H~ /\ y = {x e. H~ | (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))})
24	23, 3, 6	copab 2666	. 2 class {<.t, y>. | (t:H~-->H~ /\ y = {x e. H~ | (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))})}
25	1, 24	wceq 956	1 wff eigvec = {<.t, y>. | (t:H~-->H~ /\ y = {x e. H~ | (x =/= 0h /\ E.z e. CC (t` x) = (z .h x))})}

This definition is referenced by:  eigvecvalt 9822

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