restrict_trim — Mathlib

English

For any scalar r, if v ⟂ᵥ w then v ⟂ᵥ r • w.

Русский

Для любого скаляра r: если v ⟂ᵥ w, то v ⟂ᵥ r • w.

LaTeX

$$$ v \\perp_{\\mathcal{V}} w \\Rightarrow v \\perp_{\\mathcal{V}} (r \\cdot w) $$$

Lean4

theorem restrict_trim (hle : m ≤ n) {i : Set α} (hi : MeasurableSet[m] i) :
    @VectorMeasure.restrict α m M _ _ (v.trim hle) i = (v.restrict i).trim hle :=
  by
  ext j hj
  rw [@restrict_apply _ m, trim_measurableSet_eq hle hj, restrict_apply, trim_measurableSet_eq]
  all_goals measurability

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