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Theorem lsatset 36172
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 3489 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6643 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lsatset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2874 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6643 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
7 lsatset.z . . . . . . . . 9 0 = (0g𝑊)
86, 7syl6eqr 2874 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
98sneqd 4552 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
105, 9difeq12d 4076 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g𝑤)}) = (𝑉 ∖ { 0 }))
11 fveq2 6643 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12 lsatset.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
1311, 12syl6eqr 2874 . . . . . . 7 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1413fveq1d 6645 . . . . . 6 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
1510, 14mpteq12dv 5124 . . . . 5 (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
1615rneqd 5781 . . . 4 (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17 df-lsatoms 36158 . . . 4 LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
1812fvexi 6657 . . . . . . 7 𝑁 ∈ V
1918rnex 7592 . . . . . 6 ran 𝑁 ∈ V
20 p0ex 5258 . . . . . 6 {∅} ∈ V
2119, 20unex 7444 . . . . 5 (ran 𝑁 ∪ {∅}) ∈ V
22 eqid 2821 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
23 fvrn0 6671 . . . . . . . 8 (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
2423a1i 11 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
2522, 24fmpti 6849 . . . . . 6 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
26 frn 6493 . . . . . 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 5199 . . . 4 ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
2916, 17, 28fvmpt 6741 . . 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, 30syl5eq 2868 1 (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3471  cdif 3907  cun 3908  wss 3910  c0 4266  {csn 4540  cmpt 5119  ran crn 5529  wf 6324  cfv 6328  Basecbs 16462  0gc0g 16692  LSpanclspn 19719  LSAtomsclsa 36156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-lsatoms 36158
This theorem is referenced by:  islsat  36173  lsatlss  36178
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