English
An additive iso between G and H can be reinterpreted as a multiplicative iso between Multiplicative G and Multiplicative H; this is a standard translation between additive and multiplicative frameworks.
Русский
Изо аддитивного типа между G и H можно переопределить как изо умножительного между Multiplicative G и Multiplicative H; это стандартное переходное преобразование между аддитивной и умножительной формalisations.
LaTeX
$$$\\alpha \\simeq_+ \\beta \\iff (\\text{Multiplicative} \\alpha \\simeq \\text{Multiplicative} \\beta)$$$
Lean4
/-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def toMultiplicative [AddZeroClass G] [AddZeroClass H] : G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H)
where
toFun
f :=
{ toFun := AddMonoidHom.toMultiplicative f.toAddMonoidHom
invFun := AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_mul' := map_add f }
invFun
f :=
{ toFun := AddMonoidHom.toMultiplicative.symm f.toMonoidHom
invFun := AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_add' := map_mul f }
