Definition df-lpolN 38777

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 38776	. 2 class LPol
2		vw	. . 3 setvar 𝑤
3		cvv 3441	. . 3 class V
4	2	cv 1537	. . . . . . . 8 class 𝑤
5		cbs 16475	. . . . . . . 8 class Base
6	4, 5	cfv 6324	. . . . . . 7 class (Base‘𝑤)
7		vo	. . . . . . . 8 setvar 𝑜
8	7	cv 1537	. . . . . . 7 class 𝑜
9	6, 8	cfv 6324	. . . . . 6 class (𝑜‘(Base‘𝑤))
10		c0g 16705	. . . . . . . 8 class 0g
11	4, 10	cfv 6324	. . . . . . 7 class (0g‘𝑤)
12	11	csn 4525	. . . . . 6 class {(0g‘𝑤)}
13	9, 12	wceq 1538	. . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g‘𝑤)}
14		vx	. . . . . . . . . . 11 setvar 𝑥
15	14	cv 1537	. . . . . . . . . 10 class 𝑥
16	15, 6	wss 3881	. . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17		vy	. . . . . . . . . . 11 setvar 𝑦
18	17	cv 1537	. . . . . . . . . 10 class 𝑦
19	18, 6	wss 3881	. . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
20	15, 18	wss 3881	. . . . . . . . 9 wff 𝑥 ⊆ 𝑦
21	16, 19, 20	w3a 1084	. . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦)
22	18, 8	cfv 6324	. . . . . . . . 9 class (𝑜‘𝑦)
23	15, 8	cfv 6324	. . . . . . . . 9 class (𝑜‘𝑥)
24	22, 23	wss 3881	. . . . . . . 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 1536	. . . . . 6 wff ∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
27	26, 14	wal 1536	. . . . 5 wff ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
28		clsh 36271	. . . . . . . . 9 class LSHyp
29	4, 28	cfv 6324	. . . . . . . 8 class (LSHyp‘𝑤)
30	23, 29	wcel 2111	. . . . . . 7 wff (𝑜‘𝑥) ∈ (LSHyp‘𝑤)
31	23, 8	cfv 6324	. . . . . . . 8 class (𝑜‘(𝑜‘𝑥))
32	31, 15	wceq 1538	. . . . . . 7 wff (𝑜‘(𝑜‘𝑥)) = 𝑥
33	30, 32	wa 399	. . . . . 6 wff ((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
34		clsa 36270	. . . . . . 7 class LSAtoms
35	4, 34	cfv 6324	. . . . . 6 class (LSAtoms‘𝑤)
36	33, 14, 35	wral 3106	. . . . 5 wff ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
37	13, 27, 36	w3a 1084	. . . 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 19696	. . . . . 6 class LSubSp
39	4, 38	cfv 6324	. . . . 5 class (LSubSp‘𝑤)
40	6	cpw 4497	. . . . 5 class 𝒫 (Base‘𝑤)
41		cmap 8389	. . . . 5 class ↑m
42	39, 40, 41	co 7135	. . . 4 class ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤))
43	37, 7, 42	crab 3110	. . 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 5110	. 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 1538	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  38778