support_eq_bot_iff — Mathlib

English

For any I, the support vanishes to bottom if and only if I equals top.

Русский

Поддержка равна нулю тогда, когда I равен верхнему элементу, и наоборот.

LaTeX

$$support I = ⊥ ↔ I = ⊤$$

Lean4

@[simp]
theorem support_eq_bot_iff : support I = ⊥ ↔ I = ⊤ :=
  by
  refine ⟨fun H ↦ top_le_iff.mp fun U ↦ ?_, by simp +contextual⟩
  have := (U.2.fromSpec_image_zeroLocus _).trans_subset ((zeroLocus_inter_subset_supportSet I U).trans H.le)
  simp only [Set.subset_empty_iff, Set.image_eq_empty, Closeds.coe_bot] at this
  simp [PrimeSpectrum.zeroLocus_empty_iff_eq_top.mp this]

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