English
The map (x, y) ↦ (x, x · y) on G × G is a homeomorphism; it is a shear transformation of the product group by left-multiplying the second coordinate by the first.
Русский
Отображение (x, y) ↦ (x, x·y) на G × G является гомоморфизмом, являющимся гомоморфизмом-функцией; это сдвиг- «сдвиг» по одной координате.
LaTeX
$$$\text{Homeomorph.shearMulRight}\, G : G \times G \simeq_t G \times G$$$
Lean4
/-- The map `(x, y) ↦ (x, x * y)` as a homeomorphism. This is a shear mapping. -/
@[to_additive /-- The map `(x, y) ↦ (x, x + y)` as a homeomorphism. This is a shear mapping. -/
]
protected def shearMulRight : G × G ≃ₜ G × G :=
{
Equiv.prodShear (Equiv.refl _)
Equiv.mulLeft with
continuous_toFun := by dsimp; fun_prop
continuous_invFun := by dsimp; fun_prop }
