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Theorem lsatset 38972
Description: The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lsatset.v 𝑉 = (Base‘𝑊)
lsatset.n 𝑁 = (LSpan‘𝑊)
lsatset.z 0 = (0g𝑊)
lsatset.a 𝐴 = (LSAtoms‘𝑊)
Assertion
Ref Expression
lsatset (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
Distinct variable groups:   𝑣,𝑁   𝑣,𝑉   𝑣,𝑊   𝑣, 0   𝑣,𝑋
Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem lsatset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lsatset.a . 2 𝐴 = (LSAtoms‘𝑊)
2 elex 3499 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lsatset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6907 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
7 lsatset.z . . . . . . . . 9 0 = (0g𝑊)
86, 7eqtr4di 2793 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
98sneqd 4643 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
105, 9difeq12d 4137 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g𝑤)}) = (𝑉 ∖ { 0 }))
11 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12 lsatset.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
1311, 12eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1413fveq1d 6909 . . . . . 6 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
1510, 14mpteq12dv 5239 . . . . 5 (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
1615rneqd 5952 . . . 4 (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17 df-lsatoms 38958 . . . 4 LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
1812fvexi 6921 . . . . . . 7 𝑁 ∈ V
1918rnex 7933 . . . . . 6 ran 𝑁 ∈ V
20 p0ex 5390 . . . . . 6 {∅} ∈ V
2119, 20unex 7763 . . . . 5 (ran 𝑁 ∪ {∅}) ∈ V
22 eqid 2735 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
23 fvrn0 6937 . . . . . . . 8 (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
2423a1i 11 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
2522, 24fmpti 7132 . . . . . 6 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
26 frn 6744 . . . . . 6 ((𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅}) → ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ⊆ (ran 𝑁 ∪ {∅}))
2725, 26ax-mp 5 . . . . 5 ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ⊆ (ran 𝑁 ∪ {∅})
2821, 27ssexi 5328 . . . 4 ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
2916, 17, 28fvmpt 7016 . . 3 (𝑊 ∈ V → (LSAtoms‘𝑊) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
302, 29syl 17 . 2 (𝑊𝑋 → (LSAtoms‘𝑊) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
311, 30eqtrid 2787 1 (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  wss 3963  c0 4339  {csn 4631  cmpt 5231  ran crn 5690  wf 6559  cfv 6563  Basecbs 17245  0gc0g 17486  LSpanclspn 20987  LSAtomsclsa 38956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-lsatoms 38958
This theorem is referenced by:  islsat  38973  lsatlss  38978
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