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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 4146 | . 2 ⊢ (𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 | |
2 | islsati.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | islsati.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | eqid 2735 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | islsati.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | 2, 3, 4, 5 | islsat 38973 | . . 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 4065 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 ‘cfv 6563 Basecbs 17245 0gc0g 17486 LSpanclspn 20987 LSAtomsclsa 38956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-lsatoms 38958 |
This theorem is referenced by: lsmsatcv 38992 dihjat2 41414 dvh4dimlem 41426 lcfl8 41485 mapdval2N 41613 mapdspex 41651 hdmaprnlem16N 41845 |
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