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Definition df-lpolN 36589
Description: Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
Assertion
Ref Expression
df-lpolN LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
Distinct variable group:   𝑤,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-lpolN
StepHypRef Expression
1 clpoN 36588 . 2 class LPol
2 vw . . 3 setvar 𝑤
3 cvv 3195 . . 3 class V
42cv 1480 . . . . . . . 8 class 𝑤
5 cbs 15838 . . . . . . . 8 class Base
64, 5cfv 5876 . . . . . . 7 class (Base‘𝑤)
7 vo . . . . . . . 8 setvar 𝑜
87cv 1480 . . . . . . 7 class 𝑜
96, 8cfv 5876 . . . . . 6 class (𝑜‘(Base‘𝑤))
10 c0g 16081 . . . . . . . 8 class 0g
114, 10cfv 5876 . . . . . . 7 class (0g𝑤)
1211csn 4168 . . . . . 6 class {(0g𝑤)}
139, 12wceq 1481 . . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g𝑤)}
14 vx . . . . . . . . . . 11 setvar 𝑥
1514cv 1480 . . . . . . . . . 10 class 𝑥
1615, 6wss 3567 . . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17 vy . . . . . . . . . . 11 setvar 𝑦
1817cv 1480 . . . . . . . . . 10 class 𝑦
1918, 6wss 3567 . . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
2015, 18wss 3567 . . . . . . . . 9 wff 𝑥𝑦
2116, 19, 20w3a 1036 . . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦)
2218, 8cfv 5876 . . . . . . . . 9 class (𝑜𝑦)
2315, 8cfv 5876 . . . . . . . . 9 class (𝑜𝑥)
2422, 23wss 3567 . . . . . . . 8 wff (𝑜𝑦) ⊆ (𝑜𝑥)
2521, 24wi 4 . . . . . . 7 wff ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2625, 17wal 1479 . . . . . 6 wff 𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2726, 14wal 1479 . . . . 5 wff 𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
28 clsh 34081 . . . . . . . . 9 class LSHyp
294, 28cfv 5876 . . . . . . . 8 class (LSHyp‘𝑤)
3023, 29wcel 1988 . . . . . . 7 wff (𝑜𝑥) ∈ (LSHyp‘𝑤)
3123, 8cfv 5876 . . . . . . . 8 class (𝑜‘(𝑜𝑥))
3231, 15wceq 1481 . . . . . . 7 wff (𝑜‘(𝑜𝑥)) = 𝑥
3330, 32wa 384 . . . . . 6 wff ((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
34 clsa 34080 . . . . . . 7 class LSAtoms
354, 34cfv 5876 . . . . . 6 class (LSAtoms‘𝑤)
3633, 14, 35wral 2909 . . . . 5 wff 𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
3713, 27, 36w3a 1036 . . . 4 wff ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))
38 clss 18913 . . . . . 6 class LSubSp
394, 38cfv 5876 . . . . 5 class (LSubSp‘𝑤)
406cpw 4149 . . . . 5 class 𝒫 (Base‘𝑤)
41 cmap 7842 . . . . 5 class 𝑚
4239, 40, 41co 6635 . . . 4 class ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤))
4337, 7, 42crab 2913 . . 3 class {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))}
442, 3, 43cmpt 4720 . 2 class (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
451, 44wceq 1481 1 wff LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})
Colors of variables: wff setvar class
This definition is referenced by:  lpolsetN  36590
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