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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 4127 | . 2 ⊢ (𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 | |
2 | islsati.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | islsati.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | eqid 2727 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | islsati.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | 2, 3, 4, 5 | islsat 38387 | . . 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 4047 | . 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 1534 ∈ wcel 2099 ∃wrex 3065 ∖ cdif 3941 ⊆ wss 3944 {csn 4624 ‘cfv 6542 Basecbs 17165 0gc0g 17406 LSpanclspn 20837 LSAtomsclsa 38370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-lsatoms 38372 |
This theorem is referenced by: lsmsatcv 38406 dihjat2 40828 dvh4dimlem 40840 lcfl8 40899 mapdval2N 41027 mapdspex 41065 hdmaprnlem16N 41259 |
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