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Theorem linindsi 33192
 Description: The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islininds.b
islininds.z
islininds.r Scalar
islininds.e
islininds.0
Assertion
Ref Expression
linindsi linIndS finSupp linC
Distinct variable groups:   ,   ,,   ,,
Allowed substitution hints:   (,)   (,)   ()   (,)   (,)

Proof of Theorem linindsi
StepHypRef Expression
1 linindsv 33190 . . 3 linIndS
2 islininds.b . . . 4
3 islininds.z . . . 4
4 islininds.r . . . 4 Scalar
5 islininds.e . . . 4
6 islininds.0 . . . 4
72, 3, 4, 5, 6islininds 33191 . . 3 linIndS finSupp linC
81, 7syl 16 . 2 linIndS linIndS finSupp linC
98ibi 241 1 linIndS finSupp linC
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807  cvv 3109  cpw 4015   class class class wbr 4456  cfv 5594  (class class class)co 6296   cmap 7438   finSupp cfsupp 7847  cbs 14644  Scalarcsca 14715  c0g 14857   linC clinc 33149   linIndS clininds 33185 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-iota 5557  df-fv 5602  df-ov 6299  df-lininds 33187 This theorem is referenced by:  linindslinci  33193  linindscl  33196
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