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Theorem islsati 33299
 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.)
Hypotheses
Ref Expression
islsati.v 𝑉 = (Base‘𝑊)
islsati.n 𝑁 = (LSpan‘𝑊)
islsati.a 𝐴 = (LSAtoms‘𝑊)
Assertion
Ref Expression
islsati ((𝑊𝑋𝑈𝐴) → ∃𝑣𝑉 𝑈 = (𝑁‘{𝑣}))
Distinct variable groups:   𝑣,𝑁   𝑣,𝑈   𝑣,𝑉   𝑣,𝑊   𝑣,𝑋
Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem islsati
StepHypRef Expression
1 difss 3699 . 2 (𝑉 ∖ {(0g𝑊)}) ⊆ 𝑉
2 islsati.v . . . 4 𝑉 = (Base‘𝑊)
3 islsati.n . . . 4 𝑁 = (LSpan‘𝑊)
4 eqid 2610 . . . 4 (0g𝑊) = (0g𝑊)
5 islsati.a . . . 4 𝐴 = (LSAtoms‘𝑊)
62, 3, 4, 5islsat 33296 . . 3 (𝑊𝑋 → (𝑈𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ {(0g𝑊)})𝑈 = (𝑁‘{𝑣})))
76biimpa 500 . 2 ((𝑊𝑋𝑈𝐴) → ∃𝑣 ∈ (𝑉 ∖ {(0g𝑊)})𝑈 = (𝑁‘{𝑣}))
8 ssrexv 3630 . 2 ((𝑉 ∖ {(0g𝑊)}) ⊆ 𝑉 → (∃𝑣 ∈ (𝑉 ∖ {(0g𝑊)})𝑈 = (𝑁‘{𝑣}) → ∃𝑣𝑉 𝑈 = (𝑁‘{𝑣})))
91, 7, 8mpsyl 66 1 ((𝑊𝑋𝑈𝐴) → ∃𝑣𝑉 𝑈 = (𝑁‘{𝑣}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ‘cfv 5804  Basecbs 15695  0gc0g 15923  LSpanclspn 18792  LSAtomsclsa 33279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-lsatoms 33281 This theorem is referenced by:  lsmsatcv  33315  dihjat2  35738  dvh4dimlem  35750  lcfl8  35809  mapdval2N  35937  mapdspex  35975  hdmaprnlem16N  36172
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