Definition df-prjsp 39596

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, equivocating rational multiples of real numbers). (Contributed by BJ and Steven Nguyen, 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 39595	. 2 class ℙ𝕣𝕠𝕛
2		vv	. . 3 setvar 𝑣
3		clvec 19867	. . 3 class LVec
4		vb	. . . 4 setvar 𝑏
5	2	cv 1537	. . . . . 6 class 𝑣
6		cbs 16475	. . . . . 6 class Base
7	5, 6	cfv 6324	. . . . 5 class (Base‘𝑣)
8		c0g 16705	. . . . . . 7 class 0g
9	5, 8	cfv 6324	. . . . . 6 class (0g‘𝑣)
10	9	csn 4525	. . . . 5 class {(0g‘𝑣)}
11	7, 10	cdif 3878	. . . 4 class ((Base‘𝑣) ∖ {(0g‘𝑣)})
12	4	cv 1537	. . . . 5 class 𝑏
13		vx	. . . . . . . . 9 setvar 𝑥
14	13, 4	wel 2112	. . . . . . . 8 wff 𝑥 ∈ 𝑏
15		vy	. . . . . . . . 9 setvar 𝑦
16	15, 4	wel 2112	. . . . . . . 8 wff 𝑦 ∈ 𝑏
17	14, 16	wa 399	. . . . . . 7 wff (𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏)
18	13	cv 1537	. . . . . . . . 9 class 𝑥
19		vl	. . . . . . . . . . 11 setvar 𝑙
20	19	cv 1537	. . . . . . . . . 10 class 𝑙
21	15	cv 1537	. . . . . . . . . 10 class 𝑦
22		cvsca 16561	. . . . . . . . . . 11 class ·𝑠
23	5, 22	cfv 6324	. . . . . . . . . 10 class ( ·𝑠 ‘𝑣)
24	20, 21, 23	co 7135	. . . . . . . . 9 class (𝑙( ·𝑠 ‘𝑣)𝑦)
25	18, 24	wceq 1538	. . . . . . . 8 wff 𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)
26		csca 16560	. . . . . . . . . 10 class Scalar
27	5, 26	cfv 6324	. . . . . . . . 9 class (Scalar‘𝑣)
28	27, 6	cfv 6324	. . . . . . . 8 class (Base‘(Scalar‘𝑣))
29	25, 19, 28	wrex 3107	. . . . . . 7 wff ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)
30	17, 29	wa 399	. . . . . 6 wff ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))
31	30, 13, 15	copab 5092	. . . . 5 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}
32	12, 31	cqs 8271	. . . 4 class (𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))})
33	4, 11, 32	csb 3828	. . 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 5110	. 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 1538	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  39597