Definition df-prjcrv 42869

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 42839 and prjspvs 42849). (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 42868	. 2 class ℙ𝕣𝕠𝕛Crv
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12401	. . 3 class ℕ0
5		cfield 20663	. . 3 class Field
6		vf	. . . 4 setvar 𝑓
7		cc0 11026	. . . . . . . 8 class 0
8	2	cv 1540	. . . . . . . 8 class 𝑛
9		cfz 13423	. . . . . . . 8 class ...
10	7, 8, 9	co 7358	. . . . . . 7 class (0...𝑛)
11	3	cv 1540	. . . . . . 7 class 𝑘
12		cmhp 22072	. . . . . . 7 class mHomP
13	10, 11, 12	co 7358	. . . . . 6 class ((0...𝑛) mHomP 𝑘)
14	13	crn 5625	. . . . 5 class ran ((0...𝑛) mHomP 𝑘)
15	14	cuni 4863	. . . 4 class ∪ ran ((0...𝑛) mHomP 𝑘)
16	6	cv 1540	. . . . . . . 8 class 𝑓
17		cevl 22028	. . . . . . . . 9 class eval
18	10, 11, 17	co 7358	. . . . . . . 8 class ((0...𝑛) eval 𝑘)
19	16, 18	cfv 6492	. . . . . . 7 class (((0...𝑛) eval 𝑘)‘𝑓)
20		vp	. . . . . . . 8 setvar 𝑝
21	20	cv 1540	. . . . . . 7 class 𝑝
22	19, 21	cima 5627	. . . . . 6 class ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝)
23		c0g 17359	. . . . . . . 8 class 0g
24	11, 23	cfv 6492	. . . . . . 7 class (0g‘𝑘)
25	24	csn 4580	. . . . . 6 class {(0g‘𝑘)}
26	22, 25	wceq 1541	. . . . 5 wff ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}
27		cprjspn 42853	. . . . . 6 class ℙ𝕣𝕠𝕛n
28	8, 11, 27	co 7358	. . . . 5 class (𝑛ℙ𝕣𝕠𝕛n𝑘)
29	26, 20, 28	crab 3399	. . . 4 class {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}
30	6, 15, 29	cmpt 5179	. . 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 7360	. 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  42870