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Definition df-lpolN 37555
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 37554 . 2 class LPol
2 vw . . 3 setvar 𝑤
3 cvv 3414 . . 3 class V
42cv 1655 . . . . . . . 8 class 𝑤
5 cbs 16229 . . . . . . . 8 class Base
64, 5cfv 6127 . . . . . . 7 class (Base‘𝑤)
7 vo . . . . . . . 8 setvar 𝑜
87cv 1655 . . . . . . 7 class 𝑜
96, 8cfv 6127 . . . . . 6 class (𝑜‘(Base‘𝑤))
10 c0g 16460 . . . . . . . 8 class 0g
114, 10cfv 6127 . . . . . . 7 class (0g𝑤)
1211csn 4399 . . . . . 6 class {(0g𝑤)}
139, 12wceq 1656 . . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g𝑤)}
14 vx . . . . . . . . . . 11 setvar 𝑥
1514cv 1655 . . . . . . . . . 10 class 𝑥
1615, 6wss 3798 . . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17 vy . . . . . . . . . . 11 setvar 𝑦
1817cv 1655 . . . . . . . . . 10 class 𝑦
1918, 6wss 3798 . . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
2015, 18wss 3798 . . . . . . . . 9 wff 𝑥𝑦
2116, 19, 20w3a 1111 . . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦)
2218, 8cfv 6127 . . . . . . . . 9 class (𝑜𝑦)
2315, 8cfv 6127 . . . . . . . . 9 class (𝑜𝑥)
2422, 23wss 3798 . . . . . . . 8 wff (𝑜𝑦) ⊆ (𝑜𝑥)
2521, 24wi 4 . . . . . . 7 wff ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2625, 17wal 1654 . . . . . 6 wff 𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2726, 14wal 1654 . . . . 5 wff 𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
28 clsh 35049 . . . . . . . . 9 class LSHyp
294, 28cfv 6127 . . . . . . . 8 class (LSHyp‘𝑤)
3023, 29wcel 2164 . . . . . . 7 wff (𝑜𝑥) ∈ (LSHyp‘𝑤)
3123, 8cfv 6127 . . . . . . . 8 class (𝑜‘(𝑜𝑥))
3231, 15wceq 1656 . . . . . . 7 wff (𝑜‘(𝑜𝑥)) = 𝑥
3330, 32wa 386 . . . . . 6 wff ((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
34 clsa 35048 . . . . . . 7 class LSAtoms
354, 34cfv 6127 . . . . . 6 class (LSAtoms‘𝑤)
3633, 14, 35wral 3117 . . . . 5 wff 𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
3713, 27, 36w3a 1111 . . . 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 19295 . . . . . 6 class LSubSp
394, 38cfv 6127 . . . . 5 class (LSubSp‘𝑤)
406cpw 4380 . . . . 5 class 𝒫 (Base‘𝑤)
41 cmap 8127 . . . . 5 class 𝑚
4239, 40, 41co 6910 . . . 4 class ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤))
4337, 7, 42crab 3121 . . 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 4954 . 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 1656 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  37556
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