Theorem prjcrvfval 42586

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 7456	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4		oveq12 7457	. . . . . . . 8 ⊢ (((0...𝑛) = (0...𝑁) ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = ((0...𝑁) mHomP 𝐾))
5	3, 4	sylan 579	. . . . . . 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 2798	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = 𝐻)
8	7	rneqd 5963	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ran ((0...𝑛) mHomP 𝑘) = ran 𝐻)
9	8	unieqd 4944	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ∪ ran ((0...𝑛) mHomP 𝑘) = ∪ ran 𝐻)
10		oveq12 7457	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = (𝑁ℙ𝕣𝕠𝕛n𝐾))
11		prjcrvfval.p	. . . . . 6 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
12	10, 11	eqtr4di 2798	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = 𝑃)
13		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
14	3, 13	oveqan12d 7467	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) eval 𝑘) = 𝐸)
17	16	fveq1d 6922	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((0...𝑛) eval 𝑘)‘𝑓) = (𝐸‘𝑓))
18	17	imaeq1d 6088	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = ((𝐸‘𝑓) “ 𝑝))
19		fveq2 6920	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = (0g‘𝐾))
20		prjcrvfval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐾)
21	19, 20	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = 0 )
22	21	adantl 481	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (0g‘𝑘) = 0 )
23	22	sneqd 4660	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {(0g‘𝑘)} = { 0 })
24	18, 23	eqeq12d 2756	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)} ↔ ((𝐸‘𝑓) “ 𝑝) = { 0 }))
25	12, 24	rabeqbidv 3462	. . . 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 5257	. . 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 42585	. . 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 7482	. . . . . 6 ⊢ 𝐻 ∈ V
29	28	rnex 7950	. . . . 5 ⊢ ran 𝐻 ∈ V
30	29	uniex 7776	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
31	30	mptex 7260	. . 3 ⊢ (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}) ∈ V
32	26, 27, 31	ovmpoa 7605	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ Field) → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
33	1, 2, 32	syl2anc 583	1 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249  ran crn 5701   “ cima 5703  ‘cfv 6573  (class class class)co 7448  0cc0 11184  ℕ₀cn0 12553  ...cfz 13567  0_gc0g 17499  Fieldcfield 20752   eval cevl 22120   mHomP cmhp 22156  ℙ𝕣𝕠𝕛_ncprjspn 42569  ℙ𝕣𝕠𝕛Crvcprjcrv 42584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-prjcrv 42585

This theorem is referenced by:  prjcrvval  42587