units_smul_iff — Mathlib

English

If w is equal to v on s, then LinearIndepOn R v s is equivalent to LinearIndepOn R w s.

Русский

Если w совпадает с v на s, то LinearIndepOn R v s эквивалентно LinearIndepOn R w s.

LaTeX

$$LinearIndepOn R v s ↔ LinearIndepOn R w s$$

Lean4

@[simp]
theorem units_smul_iff (v : ι → M) (w : ι → Rˣ) : LinearIndependent R (w • v) ↔ LinearIndependent R v :=
  by
  refine ⟨fun h ↦ ?_, fun h ↦ h.units_smul w⟩
  convert h.units_smul (fun i ↦ (w i)⁻¹)
  simp [funext_iff]

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