Theorem linindscl 48313

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 2734	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2734	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
3		eqid 2734	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2734	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5		eqid 2734	. . 3 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
6	1, 2, 3, 4, 5	linindsi 48309	. 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 494	1 ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  𝒫 cpw 4573   class class class wbr 5116  ‘cfv 6527  (class class class)co 7399   ↑_m cmap 8834   finSupp cfsupp 9367  Basecbs 17213  Scalarcsca 17259  0_gc0g 17438   linC clinc 48266   linIndS clininds 48302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-xp 5657  df-rel 5658  df-iota 6480  df-fv 6535  df-ov 7402  df-lininds 48304

This theorem is referenced by: (None)