equiv — Mathlib

English

The equivalence IsComplement e is given by an explicit isomorphism G ≃ S × T with inverse given by the product, i.e., every g ∈ G decomposes uniquely as g = s t with s ∈ S, t ∈ T.

Русский

Эквиваленция IsComplement устанавливает биекцию G ≃ S × T, где единственная раскладка каждого элемента g: g = s t с s ∈ S, t ∈ T.

LaTeX

$$$G \\cong S \\times T$$$

Lean4

/-- The equivalence `G ≃ S × T`, such that the inverse is  `(*) : S × T → G` -/
noncomputable def equiv {S T : Set G} (hST : IsComplement S T) : G ≃ S × T :=
  (Equiv.ofBijective (fun x : S × T => x.1.1 * x.2.1) hST).symm

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