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Theorem linindscl 48218
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 2733 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2733 . . 3 (0g𝑀) = (0g𝑀)
3 eqid 2733 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2733 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 eqid 2733 . . 3 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
61, 2, 3, 4, 5linindsi 48214 . 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 1535  wcel 2104  wral 3057  𝒫 cpw 4604   class class class wbr 5149  cfv 6558  (class class class)co 7425  m cmap 8859   finSupp cfsupp 9393  Basecbs 17234  Scalarcsca 17290  0gc0g 17475   linC clinc 48171   linIndS clininds 48207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-xp 5689  df-rel 5690  df-iota 6510  df-fv 6566  df-ov 7428  df-lininds 48209
This theorem is referenced by: (None)
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