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Theorem eleigvec 23460
 Description: Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
eleigvec
Distinct variable groups:   ,   ,

Proof of Theorem eleigvec
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eigvecval 23399 . . 3
21eleq2d 2503 . 2
3 eldif 3330 . . . . 5
4 elch0 22756 . . . . . . 7
54necon3bbii 2632 . . . . . 6
65anbi2i 676 . . . . 5
73, 6bitri 241 . . . 4
87anbi1i 677 . . 3
9 fveq2 5728 . . . . . 6
10 oveq2 6089 . . . . . 6
119, 10eqeq12d 2450 . . . . 5
1211rexbidv 2726 . . . 4
1312elrab 3092 . . 3
14 df-3an 938 . . 3
158, 13, 143bitr4i 269 . 2
162, 15syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2599  wrex 2706  crab 2709   cdif 3317  wf 5450  cfv 5454  (class class class)co 6081  cc 8988  chil 22422   csm 22424  c0v 22427  c0h 22438  cei 22462 This theorem is referenced by:  eleigvec2  23461  eigvalcl  23464 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-hilex 22502  ax-hv0cl 22506 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-ch0 22755  df-eigvec 23356
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