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Definition df-lpolN 41464
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 41463 . 2 class LPol
2 vw . . 3 setvar 𝑤
3 cvv 3478 . . 3 class V
42cv 1536 . . . . . . . 8 class 𝑤
5 cbs 17245 . . . . . . . 8 class Base
64, 5cfv 6563 . . . . . . 7 class (Base‘𝑤)
7 vo . . . . . . . 8 setvar 𝑜
87cv 1536 . . . . . . 7 class 𝑜
96, 8cfv 6563 . . . . . 6 class (𝑜‘(Base‘𝑤))
10 c0g 17486 . . . . . . . 8 class 0g
114, 10cfv 6563 . . . . . . 7 class (0g𝑤)
1211csn 4631 . . . . . 6 class {(0g𝑤)}
139, 12wceq 1537 . . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g𝑤)}
14 vx . . . . . . . . . . 11 setvar 𝑥
1514cv 1536 . . . . . . . . . 10 class 𝑥
1615, 6wss 3963 . . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17 vy . . . . . . . . . . 11 setvar 𝑦
1817cv 1536 . . . . . . . . . 10 class 𝑦
1918, 6wss 3963 . . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
2015, 18wss 3963 . . . . . . . . 9 wff 𝑥𝑦
2116, 19, 20w3a 1086 . . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦)
2218, 8cfv 6563 . . . . . . . . 9 class (𝑜𝑦)
2315, 8cfv 6563 . . . . . . . . 9 class (𝑜𝑥)
2422, 23wss 3963 . . . . . . . 8 wff (𝑜𝑦) ⊆ (𝑜𝑥)
2521, 24wi 4 . . . . . . 7 wff ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2625, 17wal 1535 . . . . . 6 wff 𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
2726, 14wal 1535 . . . . 5 wff 𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥))
28 clsh 38957 . . . . . . . . 9 class LSHyp
294, 28cfv 6563 . . . . . . . 8 class (LSHyp‘𝑤)
3023, 29wcel 2106 . . . . . . 7 wff (𝑜𝑥) ∈ (LSHyp‘𝑤)
3123, 8cfv 6563 . . . . . . . 8 class (𝑜‘(𝑜𝑥))
3231, 15wceq 1537 . . . . . . 7 wff (𝑜‘(𝑜𝑥)) = 𝑥
3330, 32wa 395 . . . . . 6 wff ((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
34 clsa 38956 . . . . . . 7 class LSAtoms
354, 34cfv 6563 . . . . . 6 class (LSAtoms‘𝑤)
3633, 14, 35wral 3059 . . . . 5 wff 𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥)
3713, 27, 36w3a 1086 . . . 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 20947 . . . . . 6 class LSubSp
394, 38cfv 6563 . . . . 5 class (LSubSp‘𝑤)
406cpw 4605 . . . . 5 class 𝒫 (Base‘𝑤)
41 cmap 8865 . . . . 5 class m
4239, 40, 41co 7431 . . . 4 class ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤))
4337, 7, 42crab 3433 . . 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 5231 . 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 1537 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  41465
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