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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 2821 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2821 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
4 | eqid 2821 | . . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
5 | eqid 2821 | . . 3 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) | |
6 | 1, 2, 3, 4, 5 | linindsi 44496 | . 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 497 | 1 ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 𝒫 cpw 4538 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 finSupp cfsupp 8827 Basecbs 16477 Scalarcsca 16562 0gc0g 16707 linC clinc 44453 linIndS clininds 44489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-iota 6308 df-fv 6357 df-ov 7153 df-lininds 44491 |
This theorem is referenced by: (None) |
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