Theorems · Definition · group theory

Function.Injective.addCommGroup

{M₁ : Type u_1} →
  {M₂ : Type u_2} →
    [inst : Add M₁] →
      [inst_1 : Zero M₁] →
        [inst_2 : SMul ℕ M₁] →
          [inst_3 : Neg M₁] →
            [inst_4 : Sub M₁] →
              [inst_5 : SMul ℤ M₁] →
                [inst_6 : AddCommGroup M₂] →
                  (f : M₁ → M₂) →
                    Function.Injective f →
                      f 0 = 0 →
                        (∀ (x y : M₁), f (x + y) = f x + f y) →
                          (∀ (x : M₁), f (-x) = -f x) →
                            (∀ (x y : M₁), f (x - y) = f x - f y) →
                              (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) →
                                (∀ (x : M₁) (n : ℤ), f (n • x) = n • f x) → AddCommGroup M₁

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

Defined in

Mathlib.Algebra.Group.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 19 from the axioms · uses no axioms

Assumes

AddZeroSMulNegSubSMulAddCommGroup