Theorems · Definition · group theory

Function.Injective.monoid

{M₁ : Type u_1} →
  {M₂ : Type u_2} →
    [inst : Mul M₁] →
      [inst_1 : One M₁] →
        [inst_2 : Pow M₁ ℕ] →
          [inst_3 : Monoid M₂] →
            (f : M₁ → M₂) →
              Function.Injective f →
                f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → Monoid M₁

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

Defined in

Mathlib.Algebra.Group.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 12 from the axioms · uses no axioms

Assumes

MulOnePowMonoid