Definition df-prjsp 41344

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 37846. (Contributed by BJ and SN, 29-Apr-2023.)

Assertion

Ref	Expression
df-prjsp	⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))

Distinct variable group:   𝑣,𝑏,𝑥,𝑦,𝑙

Detailed syntax breakdown of Definition df-prjsp

Step	Hyp	Ref	Expression
1		cprjsp 41343	. 2 class ℙ𝕣𝕠𝕛
2		vv	. . 3 setvar 𝑣
3		clvec 20713	. . 3 class LVec
4		vb	. . . 4 setvar 𝑏
5	2	cv 1541	. . . . . 6 class 𝑣
6		cbs 17144	. . . . . 6 class Base
7	5, 6	cfv 6544	. . . . 5 class (Base‘𝑣)
8		c0g 17385	. . . . . . 7 class 0g
9	5, 8	cfv 6544	. . . . . 6 class (0g‘𝑣)
10	9	csn 4629	. . . . 5 class {(0g‘𝑣)}
11	7, 10	cdif 3946	. . . 4 class ((Base‘𝑣) ∖ {(0g‘𝑣)})
12	4	cv 1541	. . . . 5 class 𝑏
13		vx	. . . . . . . . 9 setvar 𝑥
14	13, 4	wel 2108	. . . . . . . 8 wff 𝑥 ∈ 𝑏
15		vy	. . . . . . . . 9 setvar 𝑦
16	15, 4	wel 2108	. . . . . . . 8 wff 𝑦 ∈ 𝑏
17	14, 16	wa 397	. . . . . . 7 wff (𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏)
18	13	cv 1541	. . . . . . . . 9 class 𝑥
19		vl	. . . . . . . . . . 11 setvar 𝑙
20	19	cv 1541	. . . . . . . . . 10 class 𝑙
21	15	cv 1541	. . . . . . . . . 10 class 𝑦
22		cvsca 17201	. . . . . . . . . . 11 class ·𝑠
23	5, 22	cfv 6544	. . . . . . . . . 10 class ( ·𝑠 ‘𝑣)
24	20, 21, 23	co 7409	. . . . . . . . 9 class (𝑙( ·𝑠 ‘𝑣)𝑦)
25	18, 24	wceq 1542	. . . . . . . 8 wff 𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)
26		csca 17200	. . . . . . . . . 10 class Scalar
27	5, 26	cfv 6544	. . . . . . . . 9 class (Scalar‘𝑣)
28	27, 6	cfv 6544	. . . . . . . 8 class (Base‘(Scalar‘𝑣))
29	25, 19, 28	wrex 3071	. . . . . . 7 wff ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)
30	17, 29	wa 397	. . . . . 6 wff ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))
31	30, 13, 15	copab 5211	. . . . 5 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}
32	12, 31	cqs 8702	. . . 4 class (𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))})
33	4, 11, 32	csb 3894	. . 3 class ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))})
34	2, 3, 33	cmpt 5232	. 2 class (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
35	1, 34	wceq 1542	1 wff ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))

This definition is referenced by:  prjspval  41345