Theorem eleigvect 9876

Description: Membership in the set of eigenvectors of a Hilbert space operator.

Assertion

Ref	Expression
eleigvect	|- (T:H~-->H~ -> (A e. (eigvec` T) <-> (A e. H~ /\ A =/= 0h /\ E.x e. CC (T` A) = (x .h A))))

Distinct variable groups:   x,A   x,T

Proof of Theorem eleigvect

Step	Hyp	Ref	Expression
1		eigvecvalt 9817	. . 3 |- (T:H~-->H~ -> (eigvec` T) = {y e. H~ | (y =/= 0h /\ E.x e. CC (T` y) = (x .h y))})
2	1	eleq2d 1544	. 2 |- (T:H~-->H~ -> (A e. (eigvec` T) <-> A e. {y e. H~ | (y =/= 0h /\ E.x e. CC (T` y) = (x .h y))}))
3		neeq1 1593	. . . . 5 |- (y = A -> (y =/= 0h <-> A =/= 0h))
4		fveq2 3730	. . . . . . 7 |- (y = A -> (T` y) = (T` A))
5		opreq2 3975	. . . . . . 7 |- (y = A -> (x .h y) = (x .h A))
6	4, 5	eqeq12d 1492	. . . . . 6 |- (y = A -> ((T` y) = (x .h y) <-> (T` A) = (x .h A)))
7	6	rexbidv 1667	. . . . 5 |- (y = A -> (E.x e. CC (T` y) = (x .h y) <-> E.x e. CC (T` A) = (x .h A)))
8	3, 7	anbi12d 630	. . . 4 |- (y = A -> ((y =/= 0h /\ E.x e. CC (T` y) = (x .h y)) <-> (A =/= 0h /\ E.x e. CC (T` A) = (x .h A))))
9	8	elrab 1908	. . 3 |- (A e. {y e. H~ | (y =/= 0h /\ E.x e. CC (T` y) = (x .h y))} <-> (A e. H~ /\ (A =/= 0h /\ E.x e. CC (T` A) = (x .h A))))
10		3anass 781	. . 3 |- ((A e. H~ /\ A =/= 0h /\ E.x e. CC (T` A) = (x .h A)) <-> (A e. H~ /\ (A =/= 0h /\ E.x e. CC (T` A) = (x .h A))))
11	9, 10	bitr4 176	. 2 |- (A e. {y e. H~ | (y =/= 0h /\ E.x e. CC (T` y) = (x .h y))} <-> (A e. H~ /\ A =/= 0h /\ E.x e. CC (T` A) = (x .h A)))
12	2, 11	syl6bb 538	1 |- (T:H~-->H~ -> (A e. (eigvec` T) <-> (A e. H~ /\ A =/= 0h /\ E.x e. CC (T` A) = (x .h A))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= 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:  eleigvec2t 9877  eigvalclt 9880

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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