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| Mirrors > Home > MPE Home > Th. List > Mathboxes > islinindfiss | Structured version Visualization version GIF version | ||
| Description: The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-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 |
|---|---|
| islinindfiss | ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | islininds.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
| 2 | islininds.z | . . . . 5 ⊢ 𝑍 = (0g‘𝑀) | |
| 3 | islininds.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑀) | |
| 4 | islininds.e | . . . . 5 ⊢ 𝐸 = (Base‘𝑅) | |
| 5 | islininds.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
| 6 | 1, 2, 3, 4, 5 | islinindfis 48339 | . . . 4 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |
| 7 | 6 | ancoms 458 | . . 3 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ Fin) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |
| 8 | 7 | 3adant3 1133 | . 2 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |
| 9 | 8 | 3anibar 1330 | 1 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3060 𝒫 cpw 4598 class class class wbr 5141 ‘cfv 6559 (class class class)co 7429 ↑m cmap 8862 Fincfn 8981 Basecbs 17243 Scalarcsca 17296 0gc0g 17480 linC clinc 48294 linIndS clininds 48330 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-1o 8502 df-map 8864 df-en 8982 df-fin 8985 df-fsupp 9398 df-lininds 48332 |
| This theorem is referenced by: (None) |
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