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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 2736 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 2 | eqid 2736 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
| 3 | eqid 2736 | . . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
| 4 | eqid 2736 | . . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
| 5 | eqid 2736 | . . 3 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) | |
| 6 | 1, 2, 3, 4, 5 | linindsi 48337 | . 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)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3060 𝒫 cpw 4598 class class class wbr 5141 ‘cfv 6559 (class class class)co 7429 ↑m cmap 8862 finSupp cfsupp 9397 Basecbs 17243 Scalarcsca 17296 0gc0g 17480 linC clinc 48294 linIndS clininds 48330 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-xp 5689 df-rel 5690 df-iota 6512 df-fv 6567 df-ov 7432 df-lininds 48332 |
| This theorem is referenced by: (None) |
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