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Theorem islsati 38376
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 4126 . 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 2726 . . . 4 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 islsati.a . . . 4 𝐴 = (LSAtomsβ€˜π‘Š)
62, 3, 4, 5islsat 38373 . . 3 (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– {(0gβ€˜π‘Š)})π‘ˆ = (π‘β€˜{𝑣})))
76biimpa 476 . 2 ((π‘Š ∈ 𝑋 ∧ π‘ˆ ∈ 𝐴) β†’ βˆƒπ‘£ ∈ (𝑉 βˆ– {(0gβ€˜π‘Š)})π‘ˆ = (π‘β€˜{𝑣}))
8 ssrexv 4046 . 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βˆ– cdif 3940   βŠ† wss 3943  {csn 4623  β€˜cfv 6536  Basecbs 17150  0gc0g 17391  LSpanclspn 20815  LSAtomsclsa 38356
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-lsatoms 38358
This theorem is referenced by:  lsmsatcv  38392  dihjat2  40814  dvh4dimlem  40826  lcfl8  40885  mapdval2N  41013  mapdspex  41051  hdmaprnlem16N  41245
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