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Definition df-prjsp 41031
Description: Define the projective space function. In the bijection between 3D lines through the origin and points in the projective plane (see section comment), this is equivalent to making any two 3D points (excluding the origin) equivalent iff one is a multiple of another. This definition does not quite give all the properties needed, since the scalars of a left vector space can be "less dense" than the vectors (for example, making equivalent rational multiples of real numbers). Compare df-lsatoms 37544. (Contributed by BJ and SN, 29-Apr-2023.)
Assertion
Ref Expression
df-prjsp ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) / π‘β¦Œ(𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}))
Distinct variable group:   𝑣,𝑏,π‘₯,𝑦,𝑙

Detailed syntax breakdown of Definition df-prjsp
StepHypRef Expression
1 cprjsp 41030 . 2 class ℙ𝕣𝕠𝕛
2 vv . . 3 setvar 𝑣
3 clvec 20635 . . 3 class LVec
4 vb . . . 4 setvar 𝑏
52cv 1540 . . . . . 6 class 𝑣
6 cbs 17109 . . . . . 6 class Base
75, 6cfv 6516 . . . . 5 class (Baseβ€˜π‘£)
8 c0g 17350 . . . . . . 7 class 0g
95, 8cfv 6516 . . . . . 6 class (0gβ€˜π‘£)
109csn 4606 . . . . 5 class {(0gβ€˜π‘£)}
117, 10cdif 3925 . . . 4 class ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})
124cv 1540 . . . . 5 class 𝑏
13 vx . . . . . . . . 9 setvar π‘₯
1413, 4wel 2107 . . . . . . . 8 wff π‘₯ ∈ 𝑏
15 vy . . . . . . . . 9 setvar 𝑦
1615, 4wel 2107 . . . . . . . 8 wff 𝑦 ∈ 𝑏
1714, 16wa 396 . . . . . . 7 wff (π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏)
1813cv 1540 . . . . . . . . 9 class π‘₯
19 vl . . . . . . . . . . 11 setvar 𝑙
2019cv 1540 . . . . . . . . . 10 class 𝑙
2115cv 1540 . . . . . . . . . 10 class 𝑦
22 cvsca 17166 . . . . . . . . . . 11 class ·𝑠
235, 22cfv 6516 . . . . . . . . . 10 class ( ·𝑠 β€˜π‘£)
2420, 21, 23co 7377 . . . . . . . . 9 class (𝑙( ·𝑠 β€˜π‘£)𝑦)
2518, 24wceq 1541 . . . . . . . 8 wff π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦)
26 csca 17165 . . . . . . . . . 10 class Scalar
275, 26cfv 6516 . . . . . . . . 9 class (Scalarβ€˜π‘£)
2827, 6cfv 6516 . . . . . . . 8 class (Baseβ€˜(Scalarβ€˜π‘£))
2925, 19, 28wrex 3069 . . . . . . 7 wff βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦)
3017, 29wa 396 . . . . . 6 wff ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))
3130, 13, 15copab 5187 . . . . 5 class {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}
3212, 31cqs 8669 . . . 4 class (𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))})
334, 11, 32csb 3873 . . 3 class ⦋((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) / π‘β¦Œ(𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))})
342, 3, 33cmpt 5208 . 2 class (𝑣 ∈ LVec ↦ ⦋((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) / π‘β¦Œ(𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}))
351, 34wceq 1541 1 wff ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) / π‘β¦Œ(𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}))
Colors of variables: wff setvar class
This definition is referenced by:  prjspval  41032
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