Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-lsatoms Structured version   Visualization version   GIF version

Definition df-lsatoms 34774
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 34772 . 2 class LSAtoms
2 vw . . 3 setvar 𝑤
3 cvv 3402 . . 3 class V
4 vv . . . . 5 setvar 𝑣
52cv 1636 . . . . . . 7 class 𝑤
6 cbs 16087 . . . . . . 7 class Base
75, 6cfv 6110 . . . . . 6 class (Base‘𝑤)
8 c0g 16324 . . . . . . . 8 class 0g
95, 8cfv 6110 . . . . . . 7 class (0g𝑤)
109csn 4381 . . . . . 6 class {(0g𝑤)}
117, 10cdif 3777 . . . . 5 class ((Base‘𝑤) ∖ {(0g𝑤)})
124cv 1636 . . . . . . 7 class 𝑣
1312csn 4381 . . . . . 6 class {𝑣}
14 clspn 19197 . . . . . . 7 class LSpan
155, 14cfv 6110 . . . . . 6 class (LSpan‘𝑤)
1613, 15cfv 6110 . . . . 5 class ((LSpan‘𝑤)‘{𝑣})
174, 11, 16cmpt 4934 . . . 4 class (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))
1817crn 5325 . . 3 class ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))
192, 3, 18cmpt 4934 . 2 class (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
201, 19wceq 1637 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  34788
  Copyright terms: Public domain W3C validator