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Definition df-lpolN 38484
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 38483 . 2 class LPol
2 vw . . 3 setvar 𝑤
3 cvv 3500 . . 3 class V
42cv 1529 . . . . . . . 8 class 𝑤
5 cbs 16473 . . . . . . . 8 class Base
64, 5cfv 6352 . . . . . . 7 class (Base‘𝑤)
7 vo . . . . . . . 8 setvar 𝑜
87cv 1529 . . . . . . 7 class 𝑜
96, 8cfv 6352 . . . . . 6 class (𝑜‘(Base‘𝑤))
10 c0g 16703 . . . . . . . 8 class 0g
114, 10cfv 6352 . . . . . . 7 class (0g𝑤)
1211csn 4564 . . . . . 6 class {(0g𝑤)}
139, 12wceq 1530 . . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g𝑤)}
14 vx . . . . . . . . . . 11 setvar 𝑥
1514cv 1529 . . . . . . . . . 10 class 𝑥
1615, 6wss 3940 . . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17 vy . . . . . . . . . . 11 setvar 𝑦
1817cv 1529 . . . . . . . . . 10 class 𝑦
1918, 6wss 3940 . . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
2015, 18wss 3940 . . . . . . . . 9 wff 𝑥𝑦
2116, 19, 20w3a 1081 . . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦)
2218, 8cfv 6352 . . . . . . . . 9 class (𝑜𝑦)
2315, 8cfv 6352 . . . . . . . . 9 class (𝑜𝑥)
2422, 23wss 3940 . . . . . . . 8 wff (𝑜𝑦) ⊆ (𝑜𝑥)
2521, 24wi 4 . . . . . . 7 wff ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2625, 17wal 1528 . . . . . 6 wff 𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2726, 14wal 1528 . . . . 5 wff 𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
28 clsh 35978 . . . . . . . . 9 class LSHyp
294, 28cfv 6352 . . . . . . . 8 class (LSHyp‘𝑤)
3023, 29wcel 2107 . . . . . . 7 wff (𝑜𝑥) ∈ (LSHyp‘𝑤)
3123, 8cfv 6352 . . . . . . . 8 class (𝑜‘(𝑜𝑥))
3231, 15wceq 1530 . . . . . . 7 wff (𝑜‘(𝑜𝑥)) = 𝑥
3330, 32wa 396 . . . . . 6 wff ((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
34 clsa 35977 . . . . . . 7 class LSAtoms
354, 34cfv 6352 . . . . . 6 class (LSAtoms‘𝑤)
3633, 14, 35wral 3143 . . . . 5 wff 𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
3713, 27, 36w3a 1081 . . . 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 19623 . . . . . 6 class LSubSp
394, 38cfv 6352 . . . . 5 class (LSubSp‘𝑤)
406cpw 4542 . . . . 5 class 𝒫 (Base‘𝑤)
41 cmap 8396 . . . . 5 class m
4239, 40, 41co 7148 . . . 4 class ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤))
4337, 7, 42crab 3147 . . 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 5143 . 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 1530 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  38485
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