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| Mirrors > Home > MPE Home > Th. List > Mathboxes > islsati | Structured version Visualization version GIF version | ||
| Description: A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| islsati.v | ⊢ 𝑉 = (Base‘𝑊) | 
| islsati.n | ⊢ 𝑁 = (LSpan‘𝑊) | 
| islsati.a | ⊢ 𝐴 = (LSAtoms‘𝑊) | 
| Ref | Expression | 
|---|---|
| islsati | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | difss 4135 | . 2 ⊢ (𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 | |
| 2 | islsati.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | islsati.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 4 | eqid 2736 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 5 | islsati.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
| 6 | 2, 3, 4, 5 | islsat 38993 | . . 3 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣}))) | 
| 7 | 6 | biimpa 476 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣})) | 
| 8 | ssrexv 4052 | . 2 ⊢ ((𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 → (∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣}) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣}))) | |
| 9 | 1, 7, 8 | mpsyl 68 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ∖ cdif 3947 ⊆ wss 3950 {csn 4625 ‘cfv 6560 Basecbs 17248 0gc0g 17485 LSpanclspn 20970 LSAtomsclsa 38976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-lsatoms 38978 | 
| This theorem is referenced by: lsmsatcv 39012 dihjat2 41434 dvh4dimlem 41446 lcfl8 41505 mapdval2N 41633 mapdspex 41671 hdmaprnlem16N 41865 | 
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