English
If there exists an injective map preserving 1 and multiplication into a commutative monoid, then the domain is a commutative monoid.
Русский
Если существует инъективное отображение, сохраняющее единицу и умножение, в коммутативный моноид, то область модуля является коммутативным моноидом.
LaTeX
$$$\exists f: M_1 \to M_2\,\big(\text{Injective}(f) \land f(1)=1 \land (\forall x,y: M_1)\, f(xy)=f(x)f(y) \land (\forall x: M_1)(\forall n:\mathbb{N})\, f(x^n)=f(x)^n\big)\land \text{CommMonoid}(M_2) \Rightarrow \text{CommMonoid}(M_1)$$
Lean4
/-- A type endowed with `1` and `*` is a cancel commutative monoid if it admits an injective map
that preserves `1` and `*` to a cancel commutative monoid. See note [reducible non-instances]. -/
@[to_additive /-- 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. -/
]
protected abbrev cancelCommMonoid [CancelCommMonoid M₂] (f : M₁ → M₂) (hf : Injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) : CancelCommMonoid M₁ :=
{ hf.leftCancelSemigroup f mul, hf.commMonoid f one mul npow with }
