idealOfEmpty_eq_bot — Mathlib

English

For x ∈ X and f ∈ C(X,R), f lies in the ideal of the complement of {x} iff f(x) = 0.

Русский

Для точки x и функции f: f ∈ idealOfSet(R, {x}^c) тогда и только тогда, когда f(x)=0.

LaTeX

$$$ f \\in idealOfSet(R,\\{x\\}^c) \\iff f(x)=0 $$$

Lean4

@[simp]
theorem idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ :=
  Ideal.ext fun f => by
    simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot, DFunLike.ext_iff,
      zero_apply]

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