Definition df-prjcrv 42645

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 42615 and prjspvs 42625). (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 42644	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12528	. . 3 class ℕ0
5		cfield 20731	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11156	. . . . . . . 8 class 0
8	2	cv 1538	. . . . . . . 8 class 𝑛
9		cfz 13548	. . . . . . . 8 class ...
10	7, 8, 9	co 7432	. . . . . . 7 class (0...𝑛)
11	3	cv 1538	. . . . . . 7 class 𝑘
12		cmhp 22134	. . . . . . 7 class mHomP
13	10, 11, 12	co 7432	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5685	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4906	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1538	. . . . . . . 8 class 𝑓
17		cevl 22098	. . . . . . . . 9 class eval
18	10, 11, 17	co 7432	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6560	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1538	. . . . . . 7 class 𝑝
22	19, 21	cima 5687	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17485	. . . . . . . 8 class 0g
24	11, 23	cfv 6560	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4625	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1539	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42629	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7432	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3435	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5224	. . 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 7434	. 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 1539	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  42646