English
If there exists an injective map f from M1 with 1, *, inv, and / into a DivInvOneMonoid preserving these operations, then M1 is a DivInvOneMonoid.
Русский
Если существует инъективное отображение f: M1 → M2, сохраняющее 1, *, обратную операцию и деление, в DivInvOneMonoid, то M1 является DivInvOneMonoid.
LaTeX
$$$\exists f: M_1 \to M_2\,\big(\text{Injective}(f) \land f(1)=1 \land (\forall x,y: M_1)\, f(xy)=f(x)f(y) \land (\forall x: M_1)\, f(x^{-1})=(f(x))^{-1} \land (\forall x,y)\, f(x/y)=f(x)/f(y) \land (\forall x)(\forall n)\, f(x^n)=f(x)^n \land (\forall x)(\forall z)\, f(x^n)=f(x)^n\big) \Rightarrow \text{DivInvOneMonoid}(M_1)$$
Lean4
/-- A type endowed with `1`, `*`, `⁻¹`, and `/` is a `DivInvOneMonoid` if it admits an injective
map that preserves `1`, `*`, `⁻¹`, and `/` to a `DivInvOneMonoid`. See note
[reducible non-instances]. -/
@[to_additive /-- A type endowed with `0`, `+`, unary `-`, and binary `-` is a
`SubNegZeroMonoid` if it admits an injective map that preserves `0`, `+`, unary `-`, and binary
`-` to a `SubNegZeroMonoid`. This version takes custom `nsmul` and `zsmul` as `[SMul ℕ M₁]` and
`[SMul ℤ M₁]` arguments. -/
]
protected abbrev divInvOneMonoid [DivInvOneMonoid M₂] (f : M₁ → M₂) (hf : Injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹) (div : ∀ x y, f (x / y) = f x / f y)
(npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : DivInvOneMonoid M₁ :=
{ hf.divInvMonoid f one mul inv div npow zpow, hf.invOneClass f one inv with }
