Theorems · Definition · group theory

Function.Injective.addCancelCommMonoid

{M₁ : Type u_1} →
  {M₂ : Type u_2} →
    [inst : Add M₁] →
      [inst_1 : Zero M₁] →
        [inst_2 : SMul ℕ M₁] →
          [inst_3 : AddCancelCommMonoid M₂] →
            (f : M₁ → M₂) →
              Function.Injective f →
                f 0 = 0 →
                  (∀ (x y : M₁), f (x + y) = f x + f y) →
                    (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → AddCancelCommMonoid M₁

A type endowed with 0 and + is an additive cancel commutative monoid if it admits an injective map that preserves 0 and + to an additive cancel commutative monoid.

Defined in

Mathlib.Algebra.Group.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 14 from the axioms · uses no axioms

Assumes

AddZeroSMulAddCancelCommMonoid