Definition df-prjcrv 43620

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 43590 and prjspvs 43600). (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 43619	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12587	. . 3 class ℕ0
5		cfield 20961	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11181	. . . . . . . 8 class 0
8	2	cv 1569	. . . . . . . 8 class 𝑛
9		cfz 13620	. . . . . . . 8 class ...
10	7, 8, 9	co 7412	. . . . . . 7 class (0...𝑛)
11	3	cv 1569	. . . . . . 7 class 𝑘
12		cmhp 22434	. . . . . . 7 class mHomP
13	10, 11, 12	co 7412	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5652	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4867	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1569	. . . . . . . 8 class 𝑓
17		cevl 22362	. . . . . . . . 9 class eval
18	10, 11, 17	co 7412	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6531	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1569	. . . . . . 7 class 𝑝
22	19, 21	cima 5654	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17590	. . . . . . . 8 class 0g
24	11, 23	cfv 6531	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4584	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1570	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 43604	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7412	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3413	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5186	. . 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 used by:  prjcrvfval  43621