Theorem linindscl 49766

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 → 𝑆 ∈ 𝒫 (Base‘𝑀))

Proof of Theorem linindscl

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2763	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
3		eqid 2763	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5		eqid 2763	. . 3 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
6	1, 2, 3, 4, 5	linindsi 49762	. 2 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑆)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = (0g‘(Scalar‘𝑀)))))
7	6	simpld 500	1 ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  𝒫 cpw 4560   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872   finSupp cfsupp 9378  Basecbs 17413  Scalarcsca 17457  0_gc0g 17636   linC clinc 49719   linIndS clininds 49755

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-iota 6509  df-fv 6563  df-ov 7443  df-lininds 49757

This theorem is used by: (None)