isClosed — Mathlib

English

In a topological abelian group, a closed complemented submodule is closed.

Русский

В топологической группе коммутативной сложности замкнутый дополненный подмодуль замкнут.

LaTeX

$$$$ \text{If } p \text{ is closed complemented, then } IsClosed(p). $$$$

Lean4

protected theorem isClosed [IsTopologicalAddGroup M] [T1Space M] {p : Submodule R M} (h : ClosedComplemented p) :
    IsClosed (p : Set M) := by
  rcases h with ⟨f, hf⟩
  have : ker (ContinuousLinearMap.id R M - p.subtypeL.comp f) = p := LinearMap.ker_id_sub_eq_of_proj hf
  exact this ▸ isClosed_ker _

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