The maximal ideals of a finite product of fields `∏ Lᵢ` are exactly the kernels of the projections, and distinct projections have distinct kernels.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Algebra_AlgebraType
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheory.ArithmeticFunction.Moebius
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.RamificationInertia.Basic
import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
import Mathlib.RingTheory.Henselian
set_option autoImplicit false

open TypeDensity
open scoped TensorProduct
open Polynomial IsLocalRing

theorem TypeDensity.pi_field_maximal_ideals (ι : Type) [Finite ι] (L : ι → Type) [∀ i, Field (L i)]
    (m : Ideal (∀ i, L i)) :
    (m.IsMaximal ↔ ∃ i, m = RingHom.ker (Pi.evalRingHom L i)) ∧ ∀ i j : ι,
      RingHom.ker (Pi.evalRingHom L i) = RingHom.ker (Pi.evalRingHom L j) → i = j := by sorry