Theorem prjcrvfval 42874

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 7366	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4		oveq12 7367	. . . . . . . 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 2789	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = 𝐻)
8	7	rneqd 5887	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ran ((0...𝑛) mHomP 𝑘) = ran 𝐻)
9	8	unieqd 4876	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ∪ ran ((0...𝑛) mHomP 𝑘) = ∪ ran 𝐻)
10		oveq12 7367	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = (𝑁ℙ𝕣𝕠𝕛n𝐾))
11		prjcrvfval.p	. . . . . 6 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
12	10, 11	eqtr4di 2789	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑛ℙ𝕣𝕠𝕛n𝑘) = 𝑃)
13		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
14	3, 13	oveqan12d 7377	. . . . . . . . 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 2789	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((0...𝑛) eval 𝑘) = 𝐸)
17	16	fveq1d 6836	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((0...𝑛) eval 𝑘)‘𝑓) = (𝐸‘𝑓))
18	17	imaeq1d 6018	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = ((𝐸‘𝑓) “ 𝑝))
19		fveq2 6834	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = (0g‘𝐾))
20		prjcrvfval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐾)
21	19, 20	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = 0 )
22	21	adantl 481	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (0g‘𝑘) = 0 )
23	22	sneqd 4592	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {(0g‘𝑘)} = { 0 })
24	18, 23	eqeq12d 2752	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)} ↔ ((𝐸‘𝑓) “ 𝑝) = { 0 }))
25	12, 24	rabeqbidv 3417	. . . 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 5185	. . 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 42873	. . 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 7392	. . . . . 6 ⊢ 𝐻 ∈ V
29	28	rnex 7852	. . . . 5 ⊢ ran 𝐻 ∈ V
30	29	uniex 7686	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
31	30	mptex 7169	. . 3 ⊢ (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}) ∈ V
32	26, 27, 31	ovmpoa 7513	. 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 1541   ∈ wcel 2113  {crab 3399  {csn 4580  ∪ cuni 4863   ↦ cmpt 5179  ran crn 5625   “ cima 5627  ‘cfv 6492  (class class class)co 7358  0cc0 11026  ℕ₀cn0 12401  ...cfz 13423  0_gc0g 17359  Fieldcfield 20663   eval cevl 22028   mHomP cmhp 22072  ℙ𝕣𝕠𝕛_ncprjspn 42857  ℙ𝕣𝕠𝕛Crvcprjcrv 42872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-prjcrv 42873

This theorem is referenced by:  prjcrvval  42875