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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 44507 | . . 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 44508 | . . 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 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 𝒫 cpw 4541 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 finSupp cfsupp 8835 Basecbs 16485 Scalarcsca 16570 0gc0g 16715 linC clinc 44466 linIndS clininds 44502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-iota 6316 df-fv 6365 df-ov 7161 df-lininds 44504 |
This theorem is referenced by: linindslinci 44510 linindscl 44513 |
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