ι_range_disjoint_one — Mathlib

English

The range of ι_R is disjoint from the scalars, i.e., the intersection of the range of ι_R with the scalar submodule 1 is {0}.

Русский

Образ ι_R дисjointен с мультипликативной единицей; пересечение образа ι_R и подмодуля скаляров равно {0}.

LaTeX

$$$\\operatorname{range}(\\iota_R) \\cap 1 = \\{0\\}$$$

Lean4

/-- The generators of the tensor algebra are disjoint from its scalars. -/
theorem ι_range_disjoint_one :
    Disjoint (LinearMap.range (ι R : M →ₗ[R] TensorAlgebra R M)) (1 : Submodule R (TensorAlgebra R M)) :=
  by
  rw [Submodule.disjoint_def, Submodule.one_eq_range]
  rintro _ ⟨x, hx⟩ ⟨r, rfl⟩
  rw [Algebra.linearMap_apply, ι_eq_algebraMap_iff] at hx
  rw [hx.2, map_zero]

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