Definition df-prjcrv 42618

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 42588 and prjspvs 42598). (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 42617	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12442	. . 3 class ℕ0
5		cfield 20639	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11068	. . . . . . . 8 class 0
8	2	cv 1539	. . . . . . . 8 class 𝑛
9		cfz 13468	. . . . . . . 8 class ...
10	7, 8, 9	co 7387	. . . . . . 7 class (0...𝑛)
11	3	cv 1539	. . . . . . 7 class 𝑘
12		cmhp 22016	. . . . . . 7 class mHomP
13	10, 11, 12	co 7387	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5639	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4871	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1539	. . . . . . . 8 class 𝑓
17		cevl 21980	. . . . . . . . 9 class eval
18	10, 11, 17	co 7387	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6511	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1539	. . . . . . 7 class 𝑝
22	19, 21	cima 5641	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17402	. . . . . . . 8 class 0g
24	11, 23	cfv 6511	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4589	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1540	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42602	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7387	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3405	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5188	. . 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 7389	. 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 1540	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  42619