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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 42028 | . . 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 42029 | . . 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 255 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 finSupp cfsupp 8158 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 linC clinc 41987 linIndS clininds 42023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-iota 5768 df-fv 5812 df-ov 6552 df-lininds 42025 |
This theorem is referenced by: linindslinci 42031 linindscl 42034 |
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