Definition df-prjcrv 42591

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 42561 and prjspvs 42571). (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 42590	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12418	. . 3 class ℕ0
5		cfield 20615	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11044	. . . . . . . 8 class 0
8	2	cv 1539	. . . . . . . 8 class 𝑛
9		cfz 13444	. . . . . . . 8 class ...
10	7, 8, 9	co 7369	. . . . . . 7 class (0...𝑛)
11	3	cv 1539	. . . . . . 7 class 𝑘
12		cmhp 21992	. . . . . . 7 class mHomP
13	10, 11, 12	co 7369	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5632	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4867	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1539	. . . . . . . 8 class 𝑓
17		cevl 21956	. . . . . . . . 9 class eval
18	10, 11, 17	co 7369	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6499	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1539	. . . . . . 7 class 𝑝
22	19, 21	cima 5634	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17378	. . . . . . . 8 class 0g
24	11, 23	cfv 6499	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4585	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1540	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42575	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7369	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3402	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5183	. . 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 7371	. 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 1540	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  42592