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Theorem islsati 38498
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 = (Baseβ€˜π‘Š)
islsati.n 𝑁 = (LSpanβ€˜π‘Š)
islsati.a 𝐴 = (LSAtomsβ€˜π‘Š)
Assertion
Ref Expression
islsati ((π‘Š ∈ 𝑋 ∧ π‘ˆ ∈ 𝐴) β†’ βˆƒπ‘£ ∈ 𝑉 π‘ˆ = (π‘β€˜{𝑣}))
Distinct variable groups:   𝑣,𝑁   𝑣,π‘ˆ   𝑣,𝑉   𝑣,π‘Š   𝑣,𝑋
Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem islsati
StepHypRef Expression
1 difss 4132 . 2 (𝑉 βˆ– {(0gβ€˜π‘Š)}) βŠ† 𝑉
2 islsati.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
3 islsati.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
4 eqid 2728 . . . 4 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 islsati.a . . . 4 𝐴 = (LSAtomsβ€˜π‘Š)
62, 3, 4, 5islsat 38495 . . 3 (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– {(0gβ€˜π‘Š)})π‘ˆ = (π‘β€˜{𝑣})))
76biimpa 475 . 2 ((π‘Š ∈ 𝑋 ∧ π‘ˆ ∈ 𝐴) β†’ βˆƒπ‘£ ∈ (𝑉 βˆ– {(0gβ€˜π‘Š)})π‘ˆ = (π‘β€˜{𝑣}))
8 ssrexv 4051 . 2 ((𝑉 βˆ– {(0gβ€˜π‘Š)}) βŠ† 𝑉 β†’ (βˆƒπ‘£ ∈ (𝑉 βˆ– {(0gβ€˜π‘Š)})π‘ˆ = (π‘β€˜{𝑣}) β†’ βˆƒπ‘£ ∈ 𝑉 π‘ˆ = (π‘β€˜{𝑣})))
91, 7, 8mpsyl 68 1 ((π‘Š ∈ 𝑋 ∧ π‘ˆ ∈ 𝐴) β†’ βˆƒπ‘£ ∈ 𝑉 π‘ˆ = (π‘β€˜{𝑣}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067   βˆ– cdif 3946   βŠ† wss 3949  {csn 4632  β€˜cfv 6553  Basecbs 17187  0gc0g 17428  LSpanclspn 20862  LSAtomsclsa 38478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-lsatoms 38480
This theorem is referenced by:  lsmsatcv  38514  dihjat2  40936  dvh4dimlem  40948  lcfl8  41007  mapdval2N  41135  mapdspex  41173  hdmaprnlem16N  41367
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