Theorems · Definition · group theory

Function.Injective.divisionMonoid

{M₁ : Type u_1} →
  {M₂ : Type u_2} →
    [inst : Mul M₁] →
      [inst_1 : One M₁] →
        [inst_2 : Pow M₁ ℕ] →
          [inst_3 : Inv M₁] →
            [inst_4 : Div M₁] →
              [inst_5 : Pow M₁ ℤ] →
                [inst_6 : DivisionMonoid M₂] →
                  (f : M₁ → M₂) →
                    Function.Injective f →
                      f 1 = 1 →
                        (∀ (x y : M₁), f (x * y) = f x * f y) →
                          (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) →
                            (∀ (x y : M₁), f (x / y) = f x / f y) →
                              (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) →
                                (∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n) → DivisionMonoid M₁

A type endowed with 1, *, ⁻¹, and / is a DivisionMonoid if it admits an injective map that preserves 1, *, ⁻¹, and / to a DivisionMonoid. See note [reducible non-instances]

Defined in

Mathlib.Algebra.Group.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 18 from the axioms · uses no axioms

Assumes

MulOnePowInvDivPowDivisionMonoid