Definition df-prjcrv 41454

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 41254 and prjspvs 41434). (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 41453	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12474	. . 3 class ℕ0
5		cfield 20362	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11112	. . . . . . . 8 class 0
8	2	cv 1540	. . . . . . . 8 class 𝑛
9		cfz 13486	. . . . . . . 8 class ...
10	7, 8, 9	co 7411	. . . . . . 7 class (0...𝑛)
11	3	cv 1540	. . . . . . 7 class 𝑘
12		cmhp 21678	. . . . . . 7 class mHomP
13	10, 11, 12	co 7411	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5677	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4908	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1540	. . . . . . . 8 class 𝑓
17		cevl 21640	. . . . . . . . 9 class eval
18	10, 11, 17	co 7411	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6543	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1540	. . . . . . 7 class 𝑝
22	19, 21	cima 5679	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17387	. . . . . . . 8 class 0g
24	11, 23	cfv 6543	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4628	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1541	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 41438	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7411	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3432	. . . 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 7413	. 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 1541	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  41455