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Theorem lsatset 37481
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 = (Baseβ€˜π‘Š)
lsatset.n 𝑁 = (LSpanβ€˜π‘Š)
lsatset.z 0 = (0gβ€˜π‘Š)
lsatset.a 𝐴 = (LSAtomsβ€˜π‘Š)
Assertion
Ref Expression
lsatset (π‘Š ∈ 𝑋 β†’ 𝐴 = ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})))
Distinct variable groups:   𝑣,𝑁   𝑣,𝑉   𝑣,π‘Š   𝑣, 0   𝑣,𝑋
Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem lsatset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lsatset.a . 2 𝐴 = (LSAtomsβ€˜π‘Š)
2 elex 3466 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6847 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 lsatset.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
6 fveq2 6847 . . . . . . . . 9 (𝑀 = π‘Š β†’ (0gβ€˜π‘€) = (0gβ€˜π‘Š))
7 lsatset.z . . . . . . . . 9 0 = (0gβ€˜π‘Š)
86, 7eqtr4di 2795 . . . . . . . 8 (𝑀 = π‘Š β†’ (0gβ€˜π‘€) = 0 )
98sneqd 4603 . . . . . . 7 (𝑀 = π‘Š β†’ {(0gβ€˜π‘€)} = { 0 })
105, 9difeq12d 4088 . . . . . 6 (𝑀 = π‘Š β†’ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) = (𝑉 βˆ– { 0 }))
11 fveq2 6847 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = (LSpanβ€˜π‘Š))
12 lsatset.n . . . . . . . 8 𝑁 = (LSpanβ€˜π‘Š)
1311, 12eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = 𝑁)
1413fveq1d 6849 . . . . . 6 (𝑀 = π‘Š β†’ ((LSpanβ€˜π‘€)β€˜{𝑣}) = (π‘β€˜{𝑣}))
1510, 14mpteq12dv 5201 . . . . 5 (𝑀 = π‘Š β†’ (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣})) = (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})))
1615rneqd 5898 . . . 4 (𝑀 = π‘Š β†’ ran (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣})) = ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})))
17 df-lsatoms 37467 . . . 4 LSAtoms = (𝑀 ∈ V ↦ ran (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣})))
1812fvexi 6861 . . . . . . 7 𝑁 ∈ V
1918rnex 7854 . . . . . 6 ran 𝑁 ∈ V
20 p0ex 5344 . . . . . 6 {βˆ…} ∈ V
2119, 20unex 7685 . . . . 5 (ran 𝑁 βˆͺ {βˆ…}) ∈ V
22 eqid 2737 . . . . . . 7 (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})) = (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣}))
23 fvrn0 6877 . . . . . . . 8 (π‘β€˜{𝑣}) ∈ (ran 𝑁 βˆͺ {βˆ…})
2423a1i 11 . . . . . . 7 (𝑣 ∈ (𝑉 βˆ– { 0 }) β†’ (π‘β€˜{𝑣}) ∈ (ran 𝑁 βˆͺ {βˆ…}))
2522, 24fmpti 7065 . . . . . 6 (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})):(𝑉 βˆ– { 0 })⟢(ran 𝑁 βˆͺ {βˆ…})
26 frn 6680 . . . . . 6 ((𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})):(𝑉 βˆ– { 0 })⟢(ran 𝑁 βˆͺ {βˆ…}) β†’ ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})) βŠ† (ran 𝑁 βˆͺ {βˆ…}))
2725, 26ax-mp 5 . . . . 5 ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})) βŠ† (ran 𝑁 βˆͺ {βˆ…})
2821, 27ssexi 5284 . . . 4 ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})) ∈ V
2916, 17, 28fvmpt 6953 . . 3 (π‘Š ∈ V β†’ (LSAtomsβ€˜π‘Š) = ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})))
302, 29syl 17 . 2 (π‘Š ∈ 𝑋 β†’ (LSAtomsβ€˜π‘Š) = ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})))
311, 30eqtrid 2789 1 (π‘Š ∈ 𝑋 β†’ 𝐴 = ran (𝑣 ∈ (𝑉 βˆ– { 0 }) ↦ (π‘β€˜{𝑣})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3448   βˆ– cdif 3912   βˆͺ cun 3913   βŠ† wss 3915  βˆ…c0 4287  {csn 4591   ↦ cmpt 5193  ran crn 5639  βŸΆwf 6497  β€˜cfv 6501  Basecbs 17090  0gc0g 17328  LSpanclspn 20448  LSAtomsclsa 37465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-lsatoms 37467
This theorem is referenced by:  islsat  37482  lsatlss  37487
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