Definition df-prjcrv 43484

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 43454 and prjspvs 43464). (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 43483	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12532	. . 3 class ℕ0
5		cfield 20897	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11128	. . . . . . . 8 class 0
8	2	cv 1569	. . . . . . . 8 class 𝑛
9		cfz 13565	. . . . . . . 8 class ...
10	7, 8, 9	co 7417	. . . . . . 7 class (0...𝑛)
11	3	cv 1569	. . . . . . 7 class 𝑘
12		cmhp 22367	. . . . . . 7 class mHomP
13	10, 11, 12	co 7417	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5660	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4870	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1569	. . . . . . . 8 class 𝑓
17		cevl 22295	. . . . . . . . 9 class eval
18	10, 11, 17	co 7417	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6537	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1569	. . . . . . 7 class 𝑝
22	19, 21	cima 5662	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17530	. . . . . . . 8 class 0g
24	11, 23	cfv 6537	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4587	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1570	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 43468	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7417	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3414	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5190	. . 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 7419	. 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  43485