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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 2622 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2622 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2622 | . . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
4 | eqid 2622 | . . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
5 | eqid 2622 | . . 3 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) | |
6 | 1, 2, 3, 4, 5 | linindsi 42236 | . 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 475 | 1 ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 𝒫 cpw 4158 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 finSupp cfsupp 8275 Basecbs 15857 Scalarcsca 15944 0gc0g 16100 linC clinc 42193 linIndS clininds 42229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-iota 5851 df-fv 5896 df-ov 6653 df-lininds 42231 |
This theorem is referenced by: (None) |
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