isBasis_nonneg_sections — Mathlib

English

The nonnegative sections form a basis in the (nonunital) C*-algebra context.

Русский

Нелокальные разделы, неотрицательные, образуют базис в контексте нелокальной C*-алгебры.

LaTeX

$$$$ \text{IsBasis}\bigl( (0 \le x) \wedge (\|x\| < 1)\bigr) $$$$

Lean4

/-- The sections of positive strict contractions form a filter basis. -/
theorem isBasis_nonneg_sections : IsBasis (fun x : A ↦ 0 ≤ x ∧ ‖x‖ < 1) ({x | · ≤ x})
    where
  nonempty := ⟨0, by simp⟩
  inter {x y} hx
    hy := by
    peel directedOn_nonneg_ball x (by simpa) y (by simpa) with z hz
    exact ⟨by simpa using hz.1, fun a ha ↦ ⟨hz.2.1.trans ha, hz.2.2.trans ha⟩⟩

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