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Mirrors > Home > MPE Home > Th. List > Mathboxes > linindsi | Structured version Visualization version GIF version |
Description: The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
islininds.b | ⊢ 𝐵 = (Base‘𝑀) |
islininds.z | ⊢ 𝑍 = (0g‘𝑀) |
islininds.r | ⊢ 𝑅 = (Scalar‘𝑀) |
islininds.e | ⊢ 𝐸 = (Base‘𝑅) |
islininds.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
linindsi | ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | linindsv 44494 | . . 3 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V)) | |
2 | islininds.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
3 | islininds.z | . . . 4 ⊢ 𝑍 = (0g‘𝑀) | |
4 | islininds.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑀) | |
5 | islininds.e | . . . 4 ⊢ 𝐸 = (Base‘𝑅) | |
6 | islininds.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
7 | 2, 3, 4, 5, 6 | islininds 44495 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑀 ∈ V) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝑆 linIndS 𝑀 → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |
9 | 8 | ibi 269 | 1 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 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: linindslinci 44497 linindscl 44500 |
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