Theorem linindscl 45419

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 45415	. 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 498	1 ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3054  𝒫 cpw 4503   class class class wbr 5043  ‘cfv 6369  (class class class)co 7202   ↑_m cmap 8497   finSupp cfsupp 8974  Basecbs 16684  Scalarcsca 16770  0_gc0g 16916   linC clinc 45372   linIndS clininds 45408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-xp 5546  df-rel 5547  df-iota 6327  df-fv 6377  df-ov 7205  df-lininds 45410

This theorem is referenced by: (None)