Definition df-prjcrv 43063

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 43033 and prjspvs 43043). (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 43062	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12437	. . 3 class ℕ0
5		cfield 20707	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11038	. . . . . . . 8 class 0
8	2	cv 1541	. . . . . . . 8 class 𝑛
9		cfz 13461	. . . . . . . 8 class ...
10	7, 8, 9	co 7367	. . . . . . 7 class (0...𝑛)
11	3	cv 1541	. . . . . . 7 class 𝑘
12		cmhp 22095	. . . . . . 7 class mHomP
13	10, 11, 12	co 7367	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5632	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4850	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1541	. . . . . . . 8 class 𝑓
17		cevl 22051	. . . . . . . . 9 class eval
18	10, 11, 17	co 7367	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6498	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1541	. . . . . . 7 class 𝑝
22	19, 21	cima 5634	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17402	. . . . . . . 8 class 0g
24	11, 23	cfv 6498	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4567	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1542	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 43047	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7367	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3389	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5166	. . 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 7369	. 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  43064