Theorem prjcrvfval 43064

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 7375	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4		oveq12 7376	. . . . . . . 8 ⊢ (((0...𝑛) = (0...𝑁) ∧ 𝑘 = 𝐾) → ((0...𝑛) mHomP 𝑘) = ((0...𝑁) mHomP 𝐾))
5	3, 4	sylan 581	. . . . . . 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 5893	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ran ((0...𝑛) mHomP 𝑘) = ran 𝐻)
9	8	unieqd 4863	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ∪ ran ((0...𝑛) mHomP 𝑘) = ∪ ran 𝐻)
10		oveq12 7376	. . . . . 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 7386	. . . . . . . . 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 6842	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (((0...𝑛) eval 𝑘)‘𝑓) = (𝐸‘𝑓))
18	17	imaeq1d 6024	. . . . . 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 2789	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0g‘𝑘) = 0 )
22	21	adantl 481	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (0g‘𝑘) = 0 )
23	22	sneqd 4579	. . . . . 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 3407	. . . 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 5172	. . 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 43063	. . 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 7401	. . . . . 6 ⊢ 𝐻 ∈ V
29	28	rnex 7861	. . . . 5 ⊢ ran 𝐻 ∈ V
30	29	uniex 7695	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
31	30	mptex 7178	. . 3 ⊢ (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}) ∈ V
32	26, 27, 31	ovmpoa 7522	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ Field) → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
33	1, 2, 32	syl2anc 585	1 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166  ran crn 5632   “ cima 5634  ‘cfv 6498  (class class class)co 7367  0cc0 11038  ℕ₀cn0 12437  ...cfz 13461  0_gc0g 17402  Fieldcfield 20707   eval cevl 22051   mHomP cmhp 22095  ℙ𝕣𝕠𝕛_ncprjspn 43047  ℙ𝕣𝕠𝕛Crvcprjcrv 43062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-prjcrv 43063

This theorem is referenced by:  prjcrvval  43065