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Theorem prjcrvval 42619
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 6907 . . . . 5 (𝑓 = 𝐹 → (𝐸𝑓) = (𝐸𝐹))
21imaeq1d 6079 . . . 4 (𝑓 = 𝐹 → ((𝐸𝑓) “ 𝑝) = ((𝐸𝐹) “ 𝑝))
32eqeq1d 2737 . . 3 (𝑓 = 𝐹 → (((𝐸𝑓) “ 𝑝) = { 0 } ↔ ((𝐸𝐹) “ 𝑝) = { 0 }))
43rabbidv 3441 . 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 42618 . 2 (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ran 𝐻 ↦ {𝑝𝑃 ∣ ((𝐸𝑓) “ 𝑝) = { 0 }}))
12 prjcrvval.f . 2 (𝜑𝐹 ran 𝐻)
137ovexi 7465 . . . 4 𝑃 ∈ V
1413rabex 5345 . . 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 1537  wcel 2106  {crab 3433  Vcvv 3478  {csn 4631   cuni 4912  ran crn 5690  cima 5692  cfv 6563  (class class class)co 7431  0cc0 11153  0cn0 12524  ...cfz 13544  0gc0g 17486  Fieldcfield 20747   eval cevl 22115   mHomP cmhp 22151  ℙ𝕣𝕠𝕛ncprjspn 42601  ℙ𝕣𝕠𝕛Crvcprjcrv 42616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-prjcrv 42617
This theorem is referenced by:  prjcrv0  42620
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