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Theorem linindscl 44847
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 2801 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2801 . . 3 (0g𝑀) = (0g𝑀)
3 eqid 2801 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2801 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 eqid 2801 . . 3 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
61, 2, 3, 4, 5linindsi 44843 . 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 498 1 (𝑆 linIndS 𝑀𝑆 ∈ 𝒫 (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  𝒫 cpw 4500   class class class wbr 5033  cfv 6328  (class class class)co 7139  m cmap 8393   finSupp cfsupp 8821  Basecbs 16478  Scalarcsca 16563  0gc0g 16708   linC clinc 44800   linIndS clininds 44836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-iota 6287  df-fv 6336  df-ov 7142  df-lininds 44838
This theorem is referenced by: (None)
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