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Definition df-lsatoms 33084
Description: Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
Assertion
Ref Expression
df-lsatoms LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
Distinct variable group:   𝑤,𝑣

Detailed syntax breakdown of Definition df-lsatoms
StepHypRef Expression
1 clsa 33082 . 2 class LSAtoms
2 vw . . 3 setvar 𝑤
3 cvv 3172 . . 3 class V
4 vv . . . . 5 setvar 𝑣
52cv 1473 . . . . . . 7 class 𝑤
6 cbs 15641 . . . . . . 7 class Base
75, 6cfv 5790 . . . . . 6 class (Base‘𝑤)
8 c0g 15869 . . . . . . . 8 class 0g
95, 8cfv 5790 . . . . . . 7 class (0g𝑤)
109csn 4124 . . . . . 6 class {(0g𝑤)}
117, 10cdif 3536 . . . . 5 class ((Base‘𝑤) ∖ {(0g𝑤)})
124cv 1473 . . . . . . 7 class 𝑣
1312csn 4124 . . . . . 6 class {𝑣}
14 clspn 18738 . . . . . . 7 class LSpan
155, 14cfv 5790 . . . . . 6 class (LSpan‘𝑤)
1613, 15cfv 5790 . . . . 5 class ((LSpan‘𝑤)‘{𝑣})
174, 11, 16cmpt 4637 . . . 4 class (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))
1817crn 5029 . . 3 class ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))
192, 3, 18cmpt 4637 . 2 class (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
201, 19wceq 1474 1 wff LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
Colors of variables: wff setvar class
This definition is referenced by:  lsatset  33098
  Copyright terms: Public domain W3C validator