Theorem eigvecvalt 9822

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 8869	. . 3 |- H~ e. V
2	1	rabex 2725	. 2 |- {x e. H~ | (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))} e. V
3		fveq1 3723	. . . . . 6 |- (t = T -> (t` x) = (T` x))
4	3	eqeq1d 1483	. . . . 5 |- (t = T -> ((t` x) = (y .h x) <-> (T` x) = (y .h x)))
5	4	rexbidv 1664	. . . 4 |- (t = T -> (E.y e. CC (t` x) = (y .h x) <-> E.y e. CC (T` x) = (y .h x)))
6	5	anbi2d 616	. . 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 1807	. 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 9779	. 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 4340	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 956   =/= wne 1585  E.wrex 1646  {crab 1648  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  H~chil 8788   .h csm 8790  0hc0v 8791  eigveccei 8828

This theorem is referenced by:  eleigvect 9881

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-eigvec 9779

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