Definition df-prjcrv 42628

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 42598 and prjspvs 42608). (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 42627	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12506	. . 3 class ℕ0
5		cfield 20695	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11134	. . . . . . . 8 class 0
8	2	cv 1539	. . . . . . . 8 class 𝑛
9		cfz 13529	. . . . . . . 8 class ...
10	7, 8, 9	co 7410	. . . . . . 7 class (0...𝑛)
11	3	cv 1539	. . . . . . 7 class 𝑘
12		cmhp 22072	. . . . . . 7 class mHomP
13	10, 11, 12	co 7410	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5660	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4888	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1539	. . . . . . . 8 class 𝑓
17		cevl 22036	. . . . . . . . 9 class eval
18	10, 11, 17	co 7410	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6536	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1539	. . . . . . 7 class 𝑝
22	19, 21	cima 5662	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17458	. . . . . . . 8 class 0g
24	11, 23	cfv 6536	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4606	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1540	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42612	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7410	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3420	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5206	. . 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 7412	. 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  42629