Definition df-lpolN 41974

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 41973	. 2 class LPol
2		vw	. . 3 setvar 𝑤
3		cvv 3432	. . 3 class V
4	2	cv 1546	. . . . . . . 8 class 𝑤
5		cbs 17177	. . . . . . . 8 class Base
6	4, 5	cfv 6492	. . . . . . 7 class (Base‘𝑤)
7		vo	. . . . . . . 8 setvar 𝑜
8	7	cv 1546	. . . . . . 7 class 𝑜
9	6, 8	cfv 6492	. . . . . 6 class (𝑜‘(Base‘𝑤))
10		c0g 17400	. . . . . . . 8 class 0g
11	4, 10	cfv 6492	. . . . . . 7 class (0g‘𝑤)
12	11	csn 4562	. . . . . 6 class {(0g‘𝑤)}
13	9, 12	wceq 1547	. . . . 5 wff (𝑜‘(Base‘𝑤)) = {(0g‘𝑤)}
14		vx	. . . . . . . . . . 11 setvar 𝑥
15	14	cv 1546	. . . . . . . . . 10 class 𝑥
16	15, 6	wss 3890	. . . . . . . . 9 wff 𝑥 ⊆ (Base‘𝑤)
17		vy	. . . . . . . . . . 11 setvar 𝑦
18	17	cv 1546	. . . . . . . . . 10 class 𝑦
19	18, 6	wss 3890	. . . . . . . . 9 wff 𝑦 ⊆ (Base‘𝑤)
20	15, 18	wss 3890	. . . . . . . . 9 wff 𝑥 ⊆ 𝑦
21	16, 19, 20	w3a 1092	. . . . . . . 8 wff (𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦)
22	18, 8	cfv 6492	. . . . . . . . 9 class (𝑜‘𝑦)
23	15, 8	cfv 6492	. . . . . . . . 9 class (𝑜‘𝑥)
24	22, 23	wss 3890	. . . . . . . 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 1545	. . . . . 6 wff ∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
27	26, 14	wal 1545	. . . . 5 wff ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥))
28		clsh 39468	. . . . . . . . 9 class LSHyp
29	4, 28	cfv 6492	. . . . . . . 8 class (LSHyp‘𝑤)
30	23, 29	wcel 2119	. . . . . . 7 wff (𝑜‘𝑥) ∈ (LSHyp‘𝑤)
31	23, 8	cfv 6492	. . . . . . . 8 class (𝑜‘(𝑜‘𝑥))
32	31, 15	wceq 1547	. . . . . . 7 wff (𝑜‘(𝑜‘𝑥)) = 𝑥
33	30, 32	wa 396	. . . . . 6 wff ((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
34		clsa 39467	. . . . . . 7 class LSAtoms
35	4, 34	cfv 6492	. . . . . 6 class (LSAtoms‘𝑤)
36	33, 14, 35	wral 3054	. . . . 5 wff ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)
37	13, 27, 36	w3a 1092	. . . 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 20928	. . . . . 6 class LSubSp
39	4, 38	cfv 6492	. . . . 5 class (LSubSp‘𝑤)
40	6	cpw 4536	. . . . 5 class 𝒫 (Base‘𝑤)
41		cmap 8770	. . . . 5 class ↑m
42	39, 40, 41	co 7363	. . . 4 class ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤))
43	37, 7, 42	crab 3392	. . 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 5160	. 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 1547	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  41975