Theorem eigvecvalt 9817

Description: The set of eigenvectors of a Hilbert space operator.

Assertion

Ref	Expression
eigvecvalt	|- (T:H~-->H~ -> (eigvec` T) = {x e. H~ | (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))})

Distinct variable group:   x,y,T

Proof of Theorem eigvecvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8864	. . 3 |- H~ e. V
2	1	rabex 2730	. 2 |- {x e. H~ | (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))} e. V
3		fveq1 3729	. . . . . 6 |- (t = T -> (t` x) = (T` x))
4	3	eqeq1d 1486	. . . . 5 |- (t = T -> ((t` x) = (y .h x) <-> (T` x) = (y .h x)))
5	4	rexbidv 1667	. . . 4 |- (t = T -> (E.y e. CC (t` x) = (y .h x) <-> E.y e. CC (T` x) = (y .h x)))
6	5	anbi2d 618	. . 3 |- (t = T -> ((x =/= 0h /\ E.y e. CC (t` x) = (y .h x)) <-> (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))))
7	6	rabbisdv 1810	. 2 |- (t = T -> {x e. H~ | (x =/= 0h /\ E.y e. CC (t` x) = (y .h x))} = {x e. H~ | (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))})
8		df-eigvec 9774	. 2 |- eigvec = {<.t, z>. | (t:H~-->H~ /\ z = {x e. H~ | (x =/= 0h /\ E.y e. CC (t` x) = (y .h x))})}
9	2, 1, 1, 7, 8	fvopabf4 4346	1 |- (T:H~-->H~ -> (eigvec` T) = {x e. H~ | (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   =/= wne 1588  E.wrex 1649  {crab 1651  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  H~chil 8783   .h csm 8785  0hc0v 8786  eigveccei 8823

This theorem is referenced by:  eleigvect 9876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-map 4330  df-eigvec 9774

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