Theorems · Definition · group theory

Function.Surjective.addCommMonoid

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

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

Defined in

Mathlib.Algebra.Group.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 13 from the axioms · uses propext

Assumes

AddZeroSMulAddCommMonoid