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Theorem lsatset 38594
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 3480 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6896 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lsatset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4eqtr4di 2783 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6896 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
7 lsatset.z . . . . . . . . 9 0 = (0g𝑊)
86, 7eqtr4di 2783 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = 0 )
98sneqd 4642 . . . . . . 7 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
105, 9difeq12d 4119 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g𝑤)}) = (𝑉 ∖ { 0 }))
11 fveq2 6896 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12 lsatset.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
1311, 12eqtr4di 2783 . . . . . . 7 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1413fveq1d 6898 . . . . . 6 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
1510, 14mpteq12dv 5240 . . . . 5 (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
1615rneqd 5940 . . . 4 (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17 df-lsatoms 38580 . . . 4 LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
1812fvexi 6910 . . . . . . 7 𝑁 ∈ V
1918rnex 7918 . . . . . 6 ran 𝑁 ∈ V
20 p0ex 5384 . . . . . 6 {∅} ∈ V
2119, 20unex 7749 . . . . 5 (ran 𝑁 ∪ {∅}) ∈ V
22 eqid 2725 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
23 fvrn0 6926 . . . . . . . 8 (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
2423a1i 11 . . . . . . 7 (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
2522, 24fmpti 7121 . . . . . 6 (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
26 frn 6730 . . . . . 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 5323 . . . 4 ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
2916, 17, 28fvmpt 7004 . . 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 2777 1 (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cdif 3941  cun 3942  wss 3944  c0 4322  {csn 4630  cmpt 5232  ran crn 5679  wf 6545  cfv 6549  Basecbs 17188  0gc0g 17429  LSpanclspn 20872  LSAtomsclsa 38578
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 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-lsatoms 38580
This theorem is referenced by:  islsat  38595  lsatlss  38600
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