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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 3993 | . 2 ⊢ (𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 | |
2 | islsati.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | islsati.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | eqid 2773 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | islsati.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | 2, 3, 4, 5 | islsat 35605 | . . 3 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣}))) |
7 | 6 | biimpa 469 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣})) |
8 | ssrexv 3919 | . 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 387 = wceq 1508 ∈ wcel 2051 ∃wrex 3084 ∖ cdif 3821 ⊆ wss 3824 {csn 4436 ‘cfv 6186 Basecbs 16338 0gc0g 16568 LSpanclspn 19478 LSAtomsclsa 35588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-fv 6194 df-lsatoms 35590 |
This theorem is referenced by: lsmsatcv 35624 dihjat2 38045 dvh4dimlem 38057 lcfl8 38116 mapdval2N 38244 mapdspex 38282 hdmaprnlem16N 38476 |
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