Definition df-prjcrv 43345

Description: Define the projective curve function. This takes a homogeneous polynomial and outputs the homogeneous coordinates where the polynomial evaluates to zero (the "zero set"). (In other words, scalar multiples are collapsed into the same projective point. See mhphf4 43315 and prjspvs 43325). (Contributed by SN, 23-Nov-2024.)

Assertion

Ref	Expression
df-prjcrv	⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))

Distinct variable group:   𝑘,𝑛,𝑓,𝑝

Detailed syntax breakdown of Definition df-prjcrv

Step	Hyp	Ref	Expression
1		cprjcrv 43344	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12505	. . 3 class ℕ0
5		cfield 20815	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11101	. . . . . . . 8 class 0
8	2	cv 1569	. . . . . . . 8 class 𝑛
9		cfz 13536	. . . . . . . 8 class ...
10	7, 8, 9	co 7412	. . . . . . 7 class (0...𝑛)
11	3	cv 1569	. . . . . . 7 class 𝑘
12		cmhp 22277	. . . . . . 7 class mHomP
13	10, 11, 12	co 7412	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5664	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4873	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1569	. . . . . . . 8 class 𝑓
17		cevl 22205	. . . . . . . . 9 class eval
18	10, 11, 17	co 7412	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6538	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1569	. . . . . . 7 class 𝑝
22	19, 21	cima 5666	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17493	. . . . . . . 8 class 0g
24	11, 23	cfv 6538	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4590	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1570	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 43329	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7412	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3416	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5193	. . 3 class (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}})
31	2, 3, 4, 5, 30	cmpo 7414	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
32	1, 31	wceq 1570	1 wff ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))

This definition is referenced by:  prjcrvfval  43346