Definition df-lpolN 42053

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 42052	. 2 class LPol
2		vw	. . 3 setvar 𝑤
3		cvv 3448	. . 3 class V
4	2	cv 1553	. . . . . . . 8 class 𝑤
5		cbs 17221	. . . . . . . 8 class Base
6	4, 5	cfv 6510	. . . . . . 7 class (Base‘𝑤)
7		vo	. . . . . . . 8 setvar 𝑜
8	7	cv 1553	. . . . . . 7 class 𝑜
9	6, 8	cfv 6510	. . . . . 6 class (𝑜‘(Base‘𝑤))
10		c0g 17444	. . . . . . . 8 class 0g
11	4, 10	cfv 6510	. . . . . . 7 class (0g‘𝑤)
12	11	csn 4576	. . . . . 6 class {(0g‘𝑤)}
13	9, 12	wceq 1554	. . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g‘𝑤)}
14		vx	. . . . . . . . . . 11 setvar 𝑥
15	14	cv 1553	. . . . . . . . . 10 class 𝑥
16	15, 6	wss 3899	. . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17		vy	. . . . . . . . . . 11 setvar 𝑦
18	17	cv 1553	. . . . . . . . . 10 class 𝑦
19	18, 6	wss 3899	. . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
20	15, 18	wss 3899	. . . . . . . . 9 wff 𝑥 ⊆ 𝑦
21	16, 19, 20	w3a 1095	. . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦)
22	18, 8	cfv 6510	. . . . . . . . 9 class (𝑜‘𝑦)
23	15, 8	cfv 6510	. . . . . . . . 9 class (𝑜‘𝑥)
24	22, 23	wss 3899	. . . . . . . 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 1552	. . . . . 6 wff ∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
27	26, 14	wal 1552	. . . . 5 wff ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
28		clsh 39547	. . . . . . . . 9 class LSHyp
29	4, 28	cfv 6510	. . . . . . . 8 class (LSHyp‘𝑤)
30	23, 29	wcel 2136	. . . . . . 7 wff (𝑜‘𝑥) ∈ (LSHyp‘𝑤)
31	23, 8	cfv 6510	. . . . . . . 8 class (𝑜‘(𝑜‘𝑥))
32	31, 15	wceq 1554	. . . . . . 7 wff (𝑜‘(𝑜‘𝑥)) = 𝑥
33	30, 32	wa 398	. . . . . 6 wff ((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
34		clsa 39546	. . . . . . 7 class LSAtoms
35	4, 34	cfv 6510	. . . . . 6 class (LSAtoms‘𝑤)
36	33, 14, 35	wral 3070	. . . . 5 wff ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
37	13, 27, 36	w3a 1095	. . . 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 20971	. . . . . 6 class LSubSp
39	4, 38	cfv 6510	. . . . 5 class (LSubSp‘𝑤)
40	6	cpw 4549	. . . . 5 class 𝒫 (Base‘𝑤)
41		cmap 8796	. . . . 5 class ↑m
42	39, 40, 41	co 7385	. . . 4 class ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤))
43	37, 7, 42	crab 3408	. . 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 5175	. 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 1554	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  42054