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Definition df-lsatoms 38150
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 = (𝑀 ∈ V ↦ ran (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣})))
Distinct variable group:   𝑀,𝑣

Detailed syntax breakdown of Definition df-lsatoms
StepHypRef Expression
1 clsa 38148 . 2 class LSAtoms
2 vw . . 3 setvar 𝑀
3 cvv 3473 . . 3 class V
4 vv . . . . 5 setvar 𝑣
52cv 1539 . . . . . . 7 class 𝑀
6 cbs 17149 . . . . . . 7 class Base
75, 6cfv 6543 . . . . . 6 class (Baseβ€˜π‘€)
8 c0g 17390 . . . . . . . 8 class 0g
95, 8cfv 6543 . . . . . . 7 class (0gβ€˜π‘€)
109csn 4628 . . . . . 6 class {(0gβ€˜π‘€)}
117, 10cdif 3945 . . . . 5 class ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)})
124cv 1539 . . . . . . 7 class 𝑣
1312csn 4628 . . . . . 6 class {𝑣}
14 clspn 20727 . . . . . . 7 class LSpan
155, 14cfv 6543 . . . . . 6 class (LSpanβ€˜π‘€)
1613, 15cfv 6543 . . . . 5 class ((LSpanβ€˜π‘€)β€˜{𝑣})
174, 11, 16cmpt 5231 . . . 4 class (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣}))
1817crn 5677 . . 3 class ran (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣}))
192, 3, 18cmpt 5231 . 2 class (𝑀 ∈ V ↦ ran (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣})))
201, 19wceq 1540 1 wff LSAtoms = (𝑀 ∈ V ↦ ran (𝑣 ∈ ((Baseβ€˜π‘€) βˆ– {(0gβ€˜π‘€)}) ↦ ((LSpanβ€˜π‘€)β€˜{𝑣})))
Colors of variables: wff setvar class
This definition is referenced by:  lsatset  38164
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