mem_closure_iff — Mathlib

English

For s ⊆ R, x ∈ closure s iff x ∈ AddSubmonoid.closure (Subsemigroup.closure s).

Русский

Для множества s неопределено x ∈ closure s тогда и только тогда, когда x ∈ AddSubmonoid.closure (Subsemigroup.closure s).

LaTeX

$$x \in closure s \iff x \in AddSubmonoid.closure (Subsemigroup.closure s : Set R)$$

Lean4

theorem mem_closure_iff {s : Set R} {x} : x ∈ closure s ↔ x ∈ AddSubmonoid.closure (Subsemigroup.closure s : Set R) :=
  Set.ext_iff.mp (coe_closure_eq s) x

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