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Theorem prjcrvval 42642
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 6906 . . . . 5 (𝑓 = 𝐹 → (𝐸𝑓) = (𝐸𝐹))
21imaeq1d 6077 . . . 4 (𝑓 = 𝐹 → ((𝐸𝑓) “ 𝑝) = ((𝐸𝐹) “ 𝑝))
32eqeq1d 2739 . . 3 (𝑓 = 𝐹 → (((𝐸𝑓) “ 𝑝) = { 0 } ↔ ((𝐸𝐹) “ 𝑝) = { 0 }))
43rabbidv 3444 . 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 42641 . 2 (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ran 𝐻 ↦ {𝑝𝑃 ∣ ((𝐸𝑓) “ 𝑝) = { 0 }}))
12 prjcrvval.f . 2 (𝜑𝐹 ran 𝐻)
137ovexi 7465 . . . 4 𝑃 ∈ V
1413rabex 5339 . . 3 {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }} ∈ V
1514a1i 11 . 2 (𝜑 → {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }} ∈ V)
164, 11, 12, 15fvmptd4 7040 1 (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  {csn 4626   cuni 4907  ran crn 5686  cima 5688  cfv 6561  (class class class)co 7431  0cc0 11155  0cn0 12526  ...cfz 13547  0gc0g 17484  Fieldcfield 20730   eval cevl 22097   mHomP cmhp 22133  ℙ𝕣𝕠𝕛ncprjspn 42624  ℙ𝕣𝕠𝕛Crvcprjcrv 42639
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-prjcrv 42640
This theorem is referenced by:  prjcrv0  42643
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