Theorem eigvecvalt 9739

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 8790	. . 3 |- H~ e. V
2	1	rabex 2715	. 2 |- {x e. H~ | (x =/= 0h /\ E.y e. CC (T` x) = (y .h x))} e. V
3		fveq1 3708	. . . . . 6 |- (t = T -> (t` x) = (T` x))
4	3	eqeq1d 1475	. . . . 5 |- (t = T -> ((t` x) = (y .h x) <-> (T` x) = (y .h x)))
5	4	rexbidv 1656	. . . 4 |- (t = T -> (E.y e. CC (t` x) = (y .h x) <-> E.y e. CC (T` x) = (y .h x)))
6	5	anbi2d 614	. . 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 1798	. 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 9696	. 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 4324	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 953   =/= wne 1577  E.wrex 1638  {crab 1640  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  H~chil 8727   .h csm 8729  0hc0v 8730  eigveccei 8767

This theorem is referenced by:  eleigvect 9797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-oprab 3951  df-map 4308  df-eigvec 9696

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