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Theorem prjcrvval 42121
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 = (0g𝐾)
prjcrvfval.n (𝜑𝑁 ∈ ℕ0)
prjcrvfval.k (𝜑𝐾 ∈ Field)
prjcrvval.f (𝜑𝐹 ran 𝐻)
Assertion
Ref Expression
prjcrvval (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝𝑃 ∣ ((𝐸𝐹) “ 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.
StepHypRef Expression
1 fveq2 6892 . . . . 5 (𝑓 = 𝐹 → (𝐸𝑓) = (𝐸𝐹))
21imaeq1d 6057 . . . 4 (𝑓 = 𝐹 → ((𝐸𝑓) “ 𝑝) = ((𝐸𝐹) “ 𝑝))
32eqeq1d 2727 . . 3 (𝑓 = 𝐹 → (((𝐸𝑓) “ 𝑝) = { 0 } ↔ ((𝐸𝐹) “ 𝑝) = { 0 }))
43rabbidv 3427 . 2 (𝑓 = 𝐹 → {𝑝𝑃 ∣ ((𝐸𝑓) “ 𝑝) = { 0 }} = {𝑝𝑃 ∣ ((𝐸𝐹) “ 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 = (0g𝐾)
9 prjcrvfval.n . . 3 (𝜑𝑁 ∈ ℕ0)
10 prjcrvfval.k . . 3 (𝜑𝐾 ∈ Field)
115, 6, 7, 8, 9, 10prjcrvfval 42120 . 2 (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ran 𝐻 ↦ {𝑝𝑃 ∣ ((𝐸𝑓) “ 𝑝) = { 0 }}))
12 prjcrvval.f . 2 (𝜑𝐹 ran 𝐻)
137ovexi 7450 . . . 4 𝑃 ∈ V
1413rabex 5329 . . 3 {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }} ∈ V
1514a1i 11 . 2 (𝜑 → {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }} ∈ V)
164, 11, 12, 15fvmptd4 7024 1 (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3419  Vcvv 3463  {csn 4624   cuni 4903  ran crn 5673  cima 5675  cfv 6543  (class class class)co 7416  0cc0 11138  0cn0 12502  ...cfz 13516  0gc0g 17420  Fieldcfield 20629   eval cevl 22024   mHomP cmhp 22062  ℙ𝕣𝕠𝕛ncprjspn 42103  ℙ𝕣𝕠𝕛Crvcprjcrv 42118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-prjcrv 42119
This theorem is referenced by:  prjcrv0  42122
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