injective_of_linearIndependent — Mathlib

English

In the modular group generated by S and T, a specific product relation holds: S · S · S · T · S · T · S = T⁻¹.

Русский

В модульной группе, порождаемой S и T, выполняется определённая тождество: S · S · S · T · S · T · S = T⁻¹.

LaTeX

$$$ S \\cdot S \\cdot S \\cdot T \\cdot S \\cdot T \\cdot S = T^{-1} $$$

Lean4

theorem injective_of_linearIndependent {N : Type*} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} {ι : Type*} {v : ι → M}
    (hv : Submodule.span R (.range v) = ⊤) (hli : LinearIndependent R (f ∘ v)) : Function.Injective f :=
  by
  refine (injective_iff_map_eq_zero _).mpr fun x hx ↦ ?_
  have : x ∈ Submodule.span R (.range v) := by rw [hv]; exact mem_top
  obtain ⟨c, rfl⟩ := Finsupp.mem_span_range_iff_exists_finsupp.mp this
  simp only [map_finsuppSum, map_smul] at hx
  obtain rfl := linearIndependent_iff.mp hli c hx
  simp

Похожие утверждения