Definition df-prjsp 42634

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 39014. (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 42633	. 2 class ℙ𝕣𝕠𝕛
2		vv	. . 3 setvar 𝑣
3		clvec 21034	. . 3 class LVec
4		vb	. . . 4 setvar 𝑏
5	2	cv 1540	. . . . . 6 class 𝑣
6		cbs 17117	. . . . . 6 class Base
7	5, 6	cfv 6481	. . . . 5 class (Base‘𝑣)
8		c0g 17340	. . . . . . 7 class 0g
9	5, 8	cfv 6481	. . . . . 6 class (0g‘𝑣)
10	9	csn 4576	. . . . 5 class {(0g‘𝑣)}
11	7, 10	cdif 3899	. . . 4 class ((Base‘𝑣) ∖ {(0g‘𝑣)})
12	4	cv 1540	. . . . 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 395	. . . . . . 7 wff (𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏)
18	13	cv 1540	. . . . . . . . 9 class 𝑥
19		vl	. . . . . . . . . . 11 setvar 𝑙
20	19	cv 1540	. . . . . . . . . 10 class 𝑙
21	15	cv 1540	. . . . . . . . . 10 class 𝑦
22		cvsca 17162	. . . . . . . . . . 11 class ·𝑠
23	5, 22	cfv 6481	. . . . . . . . . 10 class ( ·𝑠 ‘𝑣)
24	20, 21, 23	co 7346	. . . . . . . . 9 class (𝑙( ·𝑠 ‘𝑣)𝑦)
25	18, 24	wceq 1541	. . . . . . . 8 wff 𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)
26		csca 17161	. . . . . . . . . 10 class Scalar
27	5, 26	cfv 6481	. . . . . . . . 9 class (Scalar‘𝑣)
28	27, 6	cfv 6481	. . . . . . . 8 class (Base‘(Scalar‘𝑣))
29	25, 19, 28	wrex 3056	. . . . . . 7 wff ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)
30	17, 29	wa 395	. . . . . 6 wff ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))
31	30, 13, 15	copab 5153	. . . . 5 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}
32	12, 31	cqs 8621	. . . 4 class (𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))})
33	4, 11, 32	csb 3850	. . 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 5172	. 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 1541	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  42635