Definition df-lpolN 41483

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)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})

Distinct variable group:   𝑤,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-lpolN

Step	Hyp	Ref	Expression
1		clpoN 41482	. 2 class LPol
2		vw	. . 3 setvar 𝑤
3		cvv 3480	. . 3 class V
4	2	cv 1539	. . . . . . . 8 class 𝑤
5		cbs 17247	. . . . . . . 8 class Base
6	4, 5	cfv 6561	. . . . . . 7 class (Base‘𝑤)
7		vo	. . . . . . . 8 setvar 𝑜
8	7	cv 1539	. . . . . . 7 class 𝑜
9	6, 8	cfv 6561	. . . . . 6 class (𝑜‘(Base‘𝑤))
10		c0g 17484	. . . . . . . 8 class 0g
11	4, 10	cfv 6561	. . . . . . 7 class (0g‘𝑤)
12	11	csn 4626	. . . . . 6 class {(0g‘𝑤)}
13	9, 12	wceq 1540	. . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g‘𝑤)}
14		vx	. . . . . . . . . . 11 setvar 𝑥
15	14	cv 1539	. . . . . . . . . 10 class 𝑥
16	15, 6	wss 3951	. . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17		vy	. . . . . . . . . . 11 setvar 𝑦
18	17	cv 1539	. . . . . . . . . 10 class 𝑦
19	18, 6	wss 3951	. . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
20	15, 18	wss 3951	. . . . . . . . 9 wff 𝑥 ⊆ 𝑦
21	16, 19, 20	w3a 1087	. . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦)
22	18, 8	cfv 6561	. . . . . . . . 9 class (𝑜‘𝑦)
23	15, 8	cfv 6561	. . . . . . . . 9 class (𝑜‘𝑥)
24	22, 23	wss 3951	. . . . . . . 8 wff (𝑜‘𝑦) ⊆ (𝑜‘𝑥)
25	21, 24	wi 4	. . . . . . 7 wff ((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
26	25, 17	wal 1538	. . . . . 6 wff ∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
27	26, 14	wal 1538	. . . . 5 wff ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
28		clsh 38976	. . . . . . . . 9 class LSHyp
29	4, 28	cfv 6561	. . . . . . . 8 class (LSHyp‘𝑤)
30	23, 29	wcel 2108	. . . . . . 7 wff (𝑜‘𝑥) ∈ (LSHyp‘𝑤)
31	23, 8	cfv 6561	. . . . . . . 8 class (𝑜‘(𝑜‘𝑥))
32	31, 15	wceq 1540	. . . . . . 7 wff (𝑜‘(𝑜‘𝑥)) = 𝑥
33	30, 32	wa 395	. . . . . 6 wff ((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
34		clsa 38975	. . . . . . 7 class LSAtoms
35	4, 34	cfv 6561	. . . . . 6 class (LSAtoms‘𝑤)
36	33, 14, 35	wral 3061	. . . . 5 wff ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
37	13, 27, 36	w3a 1087	. . . 4 wff ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))
38		clss 20929	. . . . . 6 class LSubSp
39	4, 38	cfv 6561	. . . . 5 class (LSubSp‘𝑤)
40	6	cpw 4600	. . . . 5 class 𝒫 (Base‘𝑤)
41		cmap 8866	. . . . 5 class ↑m
42	39, 40, 41	co 7431	. . . 4 class ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤))
43	37, 7, 42	crab 3436	. . 3 class {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))}
44	2, 3, 43	cmpt 5225	. 2 class (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
45	1, 44	wceq 1540	1 wff LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})

This definition is referenced by:  lpolsetN  41484