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Theorem linindscl 42034
 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 2610 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2610 . . 3 (0g𝑀) = (0g𝑀)
3 eqid 2610 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2610 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 eqid 2610 . . 3 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
61, 2, 3, 4, 5linindsi 42030 . 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 474 1 (𝑆 linIndS 𝑀𝑆 ∈ 𝒫 (Base‘𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771  0gc0g 15923   linC clinc 41987   linIndS clininds 42023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552  df-lininds 42025 This theorem is referenced by: (None)
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