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Definition df-lpolN 40438
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 = (𝑀 ∈ V ↦ {π‘œ ∈ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) ∣ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))})
Distinct variable group:   𝑀,π‘œ,π‘₯,𝑦

Detailed syntax breakdown of Definition df-lpolN
StepHypRef Expression
1 clpoN 40437 . 2 class LPol
2 vw . . 3 setvar 𝑀
3 cvv 3474 . . 3 class V
42cv 1540 . . . . . . . 8 class 𝑀
5 cbs 17146 . . . . . . . 8 class Base
64, 5cfv 6543 . . . . . . 7 class (Baseβ€˜π‘€)
7 vo . . . . . . . 8 setvar π‘œ
87cv 1540 . . . . . . 7 class π‘œ
96, 8cfv 6543 . . . . . 6 class (π‘œβ€˜(Baseβ€˜π‘€))
10 c0g 17387 . . . . . . . 8 class 0g
114, 10cfv 6543 . . . . . . 7 class (0gβ€˜π‘€)
1211csn 4628 . . . . . 6 class {(0gβ€˜π‘€)}
139, 12wceq 1541 . . . . 5 wff (π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)}
14 vx . . . . . . . . . . 11 setvar π‘₯
1514cv 1540 . . . . . . . . . 10 class π‘₯
1615, 6wss 3948 . . . . . . . . 9 wff π‘₯ βŠ† (Baseβ€˜π‘€)
17 vy . . . . . . . . . . 11 setvar 𝑦
1817cv 1540 . . . . . . . . . 10 class 𝑦
1918, 6wss 3948 . . . . . . . . 9 wff 𝑦 βŠ† (Baseβ€˜π‘€)
2015, 18wss 3948 . . . . . . . . 9 wff π‘₯ βŠ† 𝑦
2116, 19, 20w3a 1087 . . . . . . . 8 wff (π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦)
2218, 8cfv 6543 . . . . . . . . 9 class (π‘œβ€˜π‘¦)
2315, 8cfv 6543 . . . . . . . . 9 class (π‘œβ€˜π‘₯)
2422, 23wss 3948 . . . . . . . 8 wff (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)
2521, 24wi 4 . . . . . . 7 wff ((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯))
2625, 17wal 1539 . . . . . 6 wff βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯))
2726, 14wal 1539 . . . . 5 wff βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯))
28 clsh 37931 . . . . . . . . 9 class LSHyp
294, 28cfv 6543 . . . . . . . 8 class (LSHypβ€˜π‘€)
3023, 29wcel 2106 . . . . . . 7 wff (π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€)
3123, 8cfv 6543 . . . . . . . 8 class (π‘œβ€˜(π‘œβ€˜π‘₯))
3231, 15wceq 1541 . . . . . . 7 wff (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯
3330, 32wa 396 . . . . . 6 wff ((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯)
34 clsa 37930 . . . . . . 7 class LSAtoms
354, 34cfv 6543 . . . . . 6 class (LSAtomsβ€˜π‘€)
3633, 14, 35wral 3061 . . . . 5 wff βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯)
3713, 27, 36w3a 1087 . . . 4 wff ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))
38 clss 20547 . . . . . 6 class LSubSp
394, 38cfv 6543 . . . . 5 class (LSubSpβ€˜π‘€)
406cpw 4602 . . . . 5 class 𝒫 (Baseβ€˜π‘€)
41 cmap 8822 . . . . 5 class ↑m
4239, 40, 41co 7411 . . . 4 class ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€))
4337, 7, 42crab 3432 . . 3 class {π‘œ ∈ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) ∣ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))}
442, 3, 43cmpt 5231 . 2 class (𝑀 ∈ V ↦ {π‘œ ∈ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) ∣ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))})
451, 44wceq 1541 1 wff LPol = (𝑀 ∈ V ↦ {π‘œ ∈ ((LSubSpβ€˜π‘€) ↑m 𝒫 (Baseβ€˜π‘€)) ∣ ((π‘œβ€˜(Baseβ€˜π‘€)) = {(0gβ€˜π‘€)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘€) ∧ 𝑦 βŠ† (Baseβ€˜π‘€) ∧ π‘₯ βŠ† 𝑦) β†’ (π‘œβ€˜π‘¦) βŠ† (π‘œβ€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘€)((π‘œβ€˜π‘₯) ∈ (LSHypβ€˜π‘€) ∧ (π‘œβ€˜(π‘œβ€˜π‘₯)) = π‘₯))})
Colors of variables: wff setvar class
This definition is referenced by:  lpolsetN  40439
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