Theorem prjcrvval 42817

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 6832	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐸‘𝑓) = (𝐸‘𝐹))
2	1	imaeq1d 6016	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐸‘𝑓) “ 𝑝) = ((𝐸‘𝐹) “ 𝑝))
3	2	eqeq1d 2736	. . 3 ⊢ (𝑓 = 𝐹 → (((𝐸‘𝑓) “ 𝑝) = { 0 } ↔ ((𝐸‘𝐹) “ 𝑝) = { 0 }))
4	3	rabbidv 3404	. 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 42816	. 2 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
12		prjcrvval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)
13	7	ovexi 7390	. . . 4 ⊢ 𝑃 ∈ V
14	13	rabex 5282	. . 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 6963	1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3397  Vcvv 3438  {csn 4578  ∪ cuni 4861  ran crn 5623   “ cima 5625  ‘cfv 6490  (class class class)co 7356  0cc0 11024  ℕ₀cn0 12399  ...cfz 13421  0_gc0g 17357  Fieldcfield 20661   eval cevl 22026   mHomP cmhp 22070  ℙ𝕣𝕠𝕛_ncprjspn 42799  ℙ𝕣𝕠𝕛Crvcprjcrv 42814

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 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-prjcrv 42815

This theorem is referenced by:  prjcrv0  42818