mem_bot — Mathlib

English

Interchange lemma about iInf again in a simplified form.

Русский

Лемма взаимодействия iInf в упрощенном виде.

LaTeX

$$$x \\in iInf S_i \\iff \\forall i, x \\in S_i$$$

Lean4

theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalStarSubalgebra R A) ↔ x = 0 :=
  show x ∈ NonUnitalAlgebra.adjoin R (∅ ∪ star ∅ : Set A) ↔ x = 0 by
    rw [Set.star_empty, Set.union_empty, NonUnitalAlgebra.adjoin_empty, NonUnitalAlgebra.mem_bot]

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