mk_one_injective_of_isScalarTower

English

Under CompatibleSMul, the left multiplication isomorphism equals the lid of the compatible SMul construction.

Русский

При наличии CompatibleSMul изоморфизм левого умножения равен lidOfCompatibleSMul.

LaTeX

$$lmulEquiv = lidOfCompatibleSMul$$

Lean4

/-- If `M` is a `B`-module that is also an `A`-module, the canonical map
`M →ₗ[A] B ⊗[A] M` is injective. -/
theorem mk_one_injective_of_isScalarTower (M : Type*) [AddCommMonoid M] [Module R M] [Module S M]
    [IsScalarTower R S M] : Function.Injective (TensorProduct.mk R S M 1) :=
  by
  apply Function.RightInverse.injective (g := LinearMap.liftBaseChange S LinearMap.id)
  intro m
  simp

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