isIrreducible_zeroLocus_iff_of_radical

English

For any radical ideal I in R, IsIrreducible (zeroLocus I) iff I is prime.

Русский

Для радикального идеала I верно: IsIrreducible(zeroLocus(I)) эквивално I является простым.

LaTeX

$$$\\mathrm{IsIrreducible}(\\mathrm{zeroLocus}(I)) \\iff I \\text{ prime}$, при условии $I$ радикал.$$

Lean4

theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical) :
    IsIrreducible (zeroLocus (I : Set R)) ↔ I.IsPrime :=
  by
  rw [Ideal.isPrime_iff, IsIrreducible]
  apply and_congr
  · rw [Set.nonempty_iff_ne_empty, Ne, zeroLocus_empty_iff_eq_top]
  · trans ∀ x y : Ideal R, Z(I) ⊆ Z(x) ∪ Z(y) → Z(I) ⊆ Z(x) ∨ Z(I) ⊆ Z(y)
    · simp_rw [isPreirreducible_iff_isClosed_union_isClosed, isClosed_iff_zeroLocus_ideal]
      constructor
      · rintro h x y
        exact h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
      · rintro h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
        exact h x y
    · simp_rw [← zeroLocus_inf, subset_zeroLocus_iff_le_vanishingIdeal, vanishingIdeal_zeroLocus_eq_radical, hI.radical]
      constructor
      · simp_rw [← SetLike.mem_coe, ← Set.singleton_subset_iff, ← Ideal.span_le, ←
          Ideal.span_singleton_mul_span_singleton]
        refine fun h x y h' => h _ _ ?_
        rw [← hI.radical_le_iff] at h' ⊢
        simpa only [Ideal.radical_inf, Ideal.radical_mul] using h'
      · simp_rw [or_iff_not_imp_left, SetLike.not_le_iff_exists]
        rintro h s t h' ⟨x, hx, hx'⟩ y hy
        exact h (h' ⟨Ideal.mul_mem_right _ _ hx, Ideal.mul_mem_left _ _ hy⟩) hx'

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