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