Theorems · Definition · commutative algebra

AddEquiv.toLinearEquiv

{R : Type u_1} →
  {M : Type u_5} →
    {M₂ : Type u_7} →
      [inst : Semiring R] →
        [inst_1 : AddCommMonoid M] →
          [inst_2 : AddCommMonoid M₂] →
            [inst_3 : Module R M] →
              [inst_4 : Module R M₂] → (e : M ≃+ M₂) → (∀ (c : R) (x : M), e (c • x) = c • e x) → M ≃ₗ[R] M₂

An additive equivalence whose underlying function preserves smul is a linear equivalence.

Defined in

Mathlib.Algebra.Module.Equiv.Basic

Cited by

3 results in Mathlib

Foundations

Depth 21 from the axioms · uses propext, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModule