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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lindepsnlininds | Structured version Visualization version GIF version | ||
| Description: A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.) |
| Ref | Expression |
|---|---|
| lindepsnlininds | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breq12 5146 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 linIndS 𝑚 ↔ 𝑆 linIndS 𝑀)) | |
| 2 | 1 | notbid 318 | . 2 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀)) |
| 3 | df-lindeps 48334 | . 2 ⊢ linDepS = {〈𝑠, 𝑚〉 ∣ ¬ 𝑠 linIndS 𝑚} | |
| 4 | 2, 3 | brabga 5537 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5141 linIndS clininds 48330 linDepS clindeps 48331 |
| 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-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-opab 5204 df-lindeps 48334 |
| This theorem is referenced by: islindeps 48343 islininds2 48374 |
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