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Theorem lsatset 33089
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 3185 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6088 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lsatset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6088 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
7 lsatset.z . . . . . . . . 9 0 = (0g𝑊)
86, 7syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
98sneqd 4137 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
105, 9difeq12d 3691 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g𝑤)}) = (𝑉 ∖ { 0 }))
11 fveq2 6088 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12 lsatset.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
1311, 12syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1413fveq1d 6090 . . . . . 6 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
1510, 14mpteq12dv 4658 . . . . 5 (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
1615rneqd 5261 . . . 4 (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17 df-lsatoms 33075 . . . 4 LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
18 fvex 6098 . . . . . . . 8 (LSpan‘𝑊) ∈ V
1912, 18eqeltri 2684 . . . . . . 7 𝑁 ∈ V
2019rnex 6970 . . . . . 6 ran 𝑁 ∈ V
21 p0ex 4774 . . . . . 6 {∅} ∈ V
2220, 21unex 6832 . . . . 5 (ran 𝑁 ∪ {∅}) ∈ V
23 eqid 2610 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
24 fvrn0 6111 . . . . . . . 8 (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
2524a1i 11 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
2623, 25fmpti 6276 . . . . . 6 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
27 frn 5952 . . . . . 6 ((𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅}) → ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ⊆ (ran 𝑁 ∪ {∅}))
2826, 27ax-mp 5 . . . . 5 ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ⊆ (ran 𝑁 ∪ {∅})
2922, 28ssexi 4726 . . . 4 ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
3016, 17, 29fvmpt 6176 . . 3 (𝑊 ∈ V → (LSAtoms‘𝑊) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
312, 30syl 17 . 2 (𝑊𝑋 → (LSAtoms‘𝑊) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
321, 31syl5eq 2656 1 (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  wss 3540  c0 3874  {csn 4125  cmpt 4638  ran crn 5029  wf 5786  cfv 5790  Basecbs 15644  0gc0g 15872  LSpanclspn 18741  LSAtomsclsa 33073
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 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
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 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-lsatoms 33075
This theorem is referenced by:  islsat  33090  lsatlss  33095
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