Theorem linindscl 47132

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 2733	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2733	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
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))
6	1, 2, 3, 4, 5	linindsi 47128	. 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 496	1 ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  𝒫 cpw 4603   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820   finSupp cfsupp 9361  Basecbs 17144  Scalarcsca 17200  0_gc0g 17385   linC clinc 47085   linIndS clininds 47121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-iota 6496  df-fv 6552  df-ov 7412  df-lininds 47123

This theorem is referenced by: (None)