zmodAddEquiv — Mathlib

English

For all x,y ∈ ZMod n, zmod n x = zmod n y iff x = y.

Русский

Для любых x,y ∈ ZMod n, zmod n x = zmod n y ⇔ x = y.

LaTeX

$$$ zmod(n)\, x = zmod(n)\, y \iff x = y $$$

Lean4

/-- `ZMod n` is (noncanonically) isomorphic to its group of characters. -/
def zmodAddEquiv : ZMod n ≃+ AddChar (ZMod n) ℂ :=
  by
  refine AddEquiv.ofBijective (circleEquivComplex.toAddMonoidHom.comp <| AddChar.toAddMonoidHom zmodHom) ?_
  rw [Fintype.bijective_iff_injective_and_card, card_eq]
  exact ⟨circleEquivComplex.injective.comp zmod_injective, rfl⟩

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