Definition df-prjcrv 43077

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 43047 and prjspvs 43057). (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 43076	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12428	. . 3 class ℕ0
5		cfield 20698	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11029	. . . . . . . 8 class 0
8	2	cv 1541	. . . . . . . 8 class 𝑛
9		cfz 13452	. . . . . . . 8 class ...
10	7, 8, 9	co 7360	. . . . . . 7 class (0...𝑛)
11	3	cv 1541	. . . . . . 7 class 𝑘
12		cmhp 22105	. . . . . . 7 class mHomP
13	10, 11, 12	co 7360	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5625	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4851	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1541	. . . . . . . 8 class 𝑓
17		cevl 22061	. . . . . . . . 9 class eval
18	10, 11, 17	co 7360	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6492	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1541	. . . . . . 7 class 𝑝
22	19, 21	cima 5627	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17393	. . . . . . . 8 class 0g
24	11, 23	cfv 6492	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4568	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1542	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 43061	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7360	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3390	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5167	. . 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 7362	. 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 1542	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  43078