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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 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 48214 | . 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 1535 ∈ wcel 2104 ∀wral 3057 𝒫 cpw 4604 class class class wbr 5149 ‘cfv 6558 (class class class)co 7425 ↑m cmap 8859 finSupp cfsupp 9393 Basecbs 17234 Scalarcsca 17290 0gc0g 17475 linC clinc 48171 linIndS clininds 48207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-xp 5689 df-rel 5690 df-iota 6510 df-fv 6566 df-ov 7428 df-lininds 48209 |
This theorem is referenced by: (None) |
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