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Theorem linindscl 47210
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 𝑀 β†’ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€))

Proof of Theorem linindscl
Dummy variables π‘₯ 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2732 . . 3 (0gβ€˜π‘€) = (0gβ€˜π‘€)
3 eqid 2732 . . 3 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
4 eqid 2732 . . 3 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
5 eqid 2732 . . 3 (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜(Scalarβ€˜π‘€))
61, 2, 3, 4, 5linindsi 47206 . 2 (𝑆 linIndS 𝑀 β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑆)((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝑓( linC β€˜π‘€)𝑆) = (0gβ€˜π‘€)) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘€)))))
76simpld 495 1 (𝑆 linIndS 𝑀 β†’ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  π’« cpw 4602   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411   ↑m cmap 8822   finSupp cfsupp 9363  Basecbs 17146  Scalarcsca 17202  0gc0g 17387   linC clinc 47163   linIndS clininds 47199
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-iota 6495  df-fv 6551  df-ov 7414  df-lininds 47201
This theorem is referenced by: (None)
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