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Theorem linindscl 48341
Description: A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
Assertion
Ref Expression
linindscl (𝑆 linIndS 𝑀𝑆 ∈ 𝒫 (Base‘𝑀))

Proof of Theorem linindscl
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2736 . . 3 (0g𝑀) = (0g𝑀)
3 eqid 2736 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2736 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 eqid 2736 . . 3 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
61, 2, 3, 4, 5linindsi 48337 . 2 (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g𝑀)) → ∀𝑥𝑆 (𝑓𝑥) = (0g‘(Scalar‘𝑀)))))
76simpld 494 1 (𝑆 linIndS 𝑀𝑆 ∈ 𝒫 (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3060  𝒫 cpw 4598   class class class wbr 5141  cfv 6559  (class class class)co 7429  m cmap 8862   finSupp cfsupp 9397  Basecbs 17243  Scalarcsca 17296  0gc0g 17480   linC clinc 48294   linIndS clininds 48330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-xp 5689  df-rel 5690  df-iota 6512  df-fv 6567  df-ov 7432  df-lininds 48332
This theorem is referenced by: (None)
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