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Theorem lindfind 27264
 Description: A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
lindfind.s
lindfind.n
lindfind.l Scalar
lindfind.z
lindfind.k
Assertion
Ref Expression
lindfind LIndF

Proof of Theorem lindfind
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 733 . 2 LIndF
2 eldifsn 3928 . . . 4
32biimpri 199 . . 3
5 simpll 732 . . . 4 LIndF LIndF
6 lindfind.l . . . . . . 7 Scalar
7 lindfind.k . . . . . . 7
86, 7elbasfv 13513 . . . . . 6
98ad2antrl 710 . . . . 5 LIndF
10 rellindf 27256 . . . . . . 7 LIndF
1110brrelexi 4919 . . . . . 6 LIndF
1211ad2antrr 708 . . . . 5 LIndF
13 eqid 2437 . . . . . 6
14 lindfind.s . . . . . 6
15 lindfind.n . . . . . 6
16 lindfind.z . . . . . 6
1713, 14, 15, 6, 7, 16islindf 27260 . . . . 5 LIndF
189, 12, 17syl2anc 644 . . . 4 LIndF LIndF
195, 18mpbid 203 . . 3 LIndF
2019simprd 451 . 2 LIndF
21 fveq2 5729 . . . . . 6
2221oveq2d 6098 . . . . 5
23 sneq 3826 . . . . . . . 8
2423difeq2d 3466 . . . . . . 7
2524imaeq2d 5204 . . . . . 6
2625fveq2d 5733 . . . . 5
2722, 26eleq12d 2505 . . . 4
2827notbid 287 . . 3
29 oveq1 6089 . . . . 5
3029eleq1d 2503 . . . 4
3130notbid 287 . . 3
3228, 31rspc2va 3060 . 2
331, 4, 20, 32syl21anc 1184 1 LIndF
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   wceq 1653   wcel 1726   wne 2600  wral 2706  cvv 2957   cdif 3318  csn 3815   class class class wbr 4213   cdm 4879  cima 4882  wf 5451  cfv 5455  (class class class)co 6082  cbs 13470  Scalarcsca 13533  cvsca 13534  c0g 13724  clspn 16048   LIndF clindf 27252 This theorem is referenced by:  lindfind2  27266  lindfrn  27269  f1lindf  27270 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-slot 13474  df-base 13475  df-lindf 27254
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