Theorem prjcrvfval 43081

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 7364	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4		oveq12 7365	. . . . . . . 8 ⊢ (((0...𝑛) = (0...𝑁) ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = ((0...𝑁) mHomP 𝐾))
5	3, 4	sylan 586	. . . . . . 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 2792	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = 𝐻)
8	7	rneqd 5880	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ran ((0...𝑛) mHomP 𝑘) = ran 𝐻)
9	8	unieqd 4851	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ∪ ran ((0...𝑛) mHomP 𝑘) = ∪ ran 𝐻)
10		oveq12 7365	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = (𝑁ℙ𝕣𝕠𝕛n𝐾))
11		prjcrvfval.p	. . . . . 6 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
12	10, 11	eqtr4di 2792	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = 𝑃)
13		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
14	3, 13	oveqan12d 7375	. . . . . . . . 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 2792	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) eval 𝑘) = 𝐸)
17	16	fveq1d 6829	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((0...𝑛) eval 𝑘)‘𝑓) = (𝐸‘𝑓))
18	17	imaeq1d 6011	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = ((𝐸‘𝑓) “ 𝑝))
19		fveq2 6827	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = (0g‘𝐾))
20		prjcrvfval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐾)
21	19, 20	eqtr4di 2792	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = 0 )
22	21	adantl 482	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (0g‘𝑘) = 0 )
23	22	sneqd 4567	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {(0g‘𝑘)} = { 0 })
24	18, 23	eqeq12d 2755	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)} ↔ ((𝐸‘𝑓) “ 𝑝) = { 0 }))
25	12, 24	rabeqbidv 3409	. . . 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 5159	. . 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 43080	. . 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 7390	. . . . . 6 ⊢ 𝐻 ∈ V
29	28	rnex 7850	. . . . 5 ⊢ ran 𝐻 ∈ V
30	29	uniex 7684	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
31	30	mptex 7167	. . 3 ⊢ (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}) ∈ V
32	26, 27, 31	ovmpoa 7511	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ Field) → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
33	1, 2, 32	syl2anc 590	1 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3391  {csn 4555  ∪ cuni 4838   ↦ cmpt 5153  ran crn 5619   “ cima 5621  ‘cfv 6485  (class class class)co 7356  0cc0 11029  ℕ₀cn0 12428  ...cfz 13452  0_gc0g 17393  Fieldcfield 20702   eval cevl 22049   mHomP cmhp 22121  ℙ𝕣𝕠𝕛_ncprjspn 43064  ℙ𝕣𝕠𝕛Crvcprjcrv 43079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-prjcrv 43080

This theorem is referenced by:  prjcrvval  43082