Definition df-prjcrv 42748

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 42718 and prjspvs 42728). (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 42747	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12388	. . 3 class ℕ0
5		cfield 20647	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11013	. . . . . . . 8 class 0
8	2	cv 1540	. . . . . . . 8 class 𝑛
9		cfz 13409	. . . . . . . 8 class ...
10	7, 8, 9	co 7352	. . . . . . 7 class (0...𝑛)
11	3	cv 1540	. . . . . . 7 class 𝑘
12		cmhp 22045	. . . . . . 7 class mHomP
13	10, 11, 12	co 7352	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5620	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4858	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1540	. . . . . . . 8 class 𝑓
17		cevl 22009	. . . . . . . . 9 class eval
18	10, 11, 17	co 7352	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6486	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1540	. . . . . . 7 class 𝑝
22	19, 21	cima 5622	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17345	. . . . . . . 8 class 0g
24	11, 23	cfv 6486	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4575	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1541	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42732	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7352	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3396	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5174	. . 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 7354	. 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 1541	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  42749