Definition df-prjcrv 42617

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 42587 and prjspvs 42597). (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 42616	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12524	. . 3 class ℕ0
5		cfield 20747	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11153	. . . . . . . 8 class 0
8	2	cv 1536	. . . . . . . 8 class 𝑛
9		cfz 13544	. . . . . . . 8 class ...
10	7, 8, 9	co 7431	. . . . . . 7 class (0...𝑛)
11	3	cv 1536	. . . . . . 7 class 𝑘
12		cmhp 22151	. . . . . . 7 class mHomP
13	10, 11, 12	co 7431	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5690	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4912	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1536	. . . . . . . 8 class 𝑓
17		cevl 22115	. . . . . . . . 9 class eval
18	10, 11, 17	co 7431	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6563	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1536	. . . . . . 7 class 𝑝
22	19, 21	cima 5692	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17486	. . . . . . . 8 class 0g
24	11, 23	cfv 6563	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4631	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1537	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42601	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7431	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3433	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5231	. . 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 7433	. 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 1537	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  42618