Theorem prjcrvfval 42592

Description: Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)

Hypotheses

Ref	Expression
prjcrvfval.h	⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)
prjcrvfval.e	⊢ 𝐸 = ((0...𝑁) eval 𝐾)
prjcrvfval.p	⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
prjcrvfval.0	⊢ 0 = (0g‘𝐾)
prjcrvfval.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
prjcrvfval.k	⊢ (𝜑 → 𝐾 ∈ Field)

Assertion

Ref	Expression
prjcrvfval	⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))

Distinct variable groups:   𝑓,𝑁,𝑝   𝑓,𝐾,𝑝   𝑃,𝑝   𝑓,𝐻

Allowed substitution hints:   𝜑(𝑓,𝑝)   𝑃(𝑓)   𝐸(𝑓,𝑝)   𝐻(𝑝)   0 (𝑓,𝑝)

Proof of Theorem prjcrvfval

Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prjcrvfval.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		prjcrvfval.k	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
3		oveq2 7377	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4		oveq12 7378	. . . . . . . 8 ⊢ (((0...𝑛) = (0...𝑁) ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = ((0...𝑁) mHomP 𝐾))
5	3, 4	sylan 580	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = ((0...𝑁) mHomP 𝐾))
6		prjcrvfval.h	. . . . . . 7 ⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)
7	5, 6	eqtr4di 2782	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = 𝐻)
8	7	rneqd 5891	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ran ((0...𝑛) mHomP 𝑘) = ran 𝐻)
9	8	unieqd 4880	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ∪ ran ((0...𝑛) mHomP 𝑘) = ∪ ran 𝐻)
10		oveq12 7378	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = (𝑁ℙ𝕣𝕠𝕛n𝐾))
11		prjcrvfval.p	. . . . . 6 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
12	10, 11	eqtr4di 2782	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = 𝑃)
13		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
14	3, 13	oveqan12d 7388	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) eval 𝑘) = ((0...𝑁) eval 𝐾))
15		prjcrvfval.e	. . . . . . . . 9 ⊢ 𝐸 = ((0...𝑁) eval 𝐾)
16	14, 15	eqtr4di 2782	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) eval 𝑘) = 𝐸)
17	16	fveq1d 6842	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((0...𝑛) eval 𝑘)‘𝑓) = (𝐸‘𝑓))
18	17	imaeq1d 6019	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = ((𝐸‘𝑓) “ 𝑝))
19		fveq2 6840	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = (0g‘𝐾))
20		prjcrvfval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐾)
21	19, 20	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = 0 )
22	21	adantl 481	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (0g‘𝑘) = 0 )
23	22	sneqd 4597	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {(0g‘𝑘)} = { 0 })
24	18, 23	eqeq12d 2745	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)} ↔ ((𝐸‘𝑓) “ 𝑝) = { 0 }))
25	12, 24	rabeqbidv 3421	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}} = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }})
26	9, 25	mpteq12dv 5189	. . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
27		df-prjcrv 42591	. . 3 ⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
28	6	ovexi 7403	. . . . . 6 ⊢ 𝐻 ∈ V
29	28	rnex 7866	. . . . 5 ⊢ ran 𝐻 ∈ V
30	29	uniex 7697	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
31	30	mptex 7179	. . 3 ⊢ (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}) ∈ V
32	26, 27, 31	ovmpoa 7524	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ Field) → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
33	1, 2, 32	syl2anc 584	1 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402  {csn 4585  ∪ cuni 4867   ↦ cmpt 5183  ran crn 5632   “ cima 5634  ‘cfv 6499  (class class class)co 7369  0cc0 11044  ℕ₀cn0 12418  ...cfz 13444  0_gc0g 17378  Fieldcfield 20615   eval cevl 21956   mHomP cmhp 21992  ℙ𝕣𝕠𝕛_ncprjspn 42575  ℙ𝕣𝕠𝕛Crvcprjcrv 42590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-prjcrv 42591

This theorem is referenced by:  prjcrvval  42593