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Mirrors > Home > MPE Home > Th. List > Mathboxes > linindscl | Structured version Visualization version GIF version |
Description: A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.) |
Ref | Expression |
---|---|
linindscl | ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀)) |
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)) |
Colors of variables: wff setvar class |
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 0gc0g 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) |
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