Theorem prjcrvval 40664

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)
prjcrvval.f	⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)

Assertion

Ref	Expression
prjcrvval	⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

Distinct variable groups:   𝑁,𝑝   𝐾,𝑝   𝑃,𝑝   𝐹,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝐸(𝑝)   𝐻(𝑝)   0 (𝑝)

Proof of Theorem prjcrvval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6804	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐸‘𝑓) = (𝐸‘𝐹))
2	1	imaeq1d 5978	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐸‘𝑓) “ 𝑝) = ((𝐸‘𝐹) “ 𝑝))
3	2	eqeq1d 2738	. . 3 ⊢ (𝑓 = 𝐹 → (((𝐸‘𝑓) “ 𝑝) = { 0 } ↔ ((𝐸‘𝐹) “ 𝑝) = { 0 }))
4	3	rabbidv 3421	. 2 ⊢ (𝑓 = 𝐹 → {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }} = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})
5		prjcrvfval.h	. . 3 ⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)
6		prjcrvfval.e	. . 3 ⊢ 𝐸 = ((0...𝑁) eval 𝐾)
7		prjcrvfval.p	. . 3 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
8		prjcrvfval.0	. . 3 ⊢ 0 = (0g‘𝐾)
9		prjcrvfval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
10		prjcrvfval.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Field)
11	5, 6, 7, 8, 9, 10	prjcrvfval 40663	. 2 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
12		prjcrvval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)
13	7	ovexi 7341	. . . 4 ⊢ 𝑃 ∈ V
14	13	rabex 5265	. . 3 ⊢ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }} ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }} ∈ V)
16	4, 11, 12, 15	fvmptd4 40405	1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  {crab 3303  Vcvv 3437  {csn 4565  ∪ cuni 4844  ran crn 5601   “ cima 5603  ‘cfv 6458  (class class class)co 7307  0cc0 10921  ℕ₀cn0 12283  ...cfz 13289  0_gc0g 17199  Fieldcfield 20041   eval cevl 21330   mHomP cmhp 21368  ℙ𝕣𝕠𝕛_ncprjspn 40648  ℙ𝕣𝕠𝕛Crvcprjcrv 40661

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-prjcrv 40662

This theorem is referenced by:  prjcrv0  40665