toAddSubmonoid — Mathlib

English

Submonoids of a monoid M are isomorphic to additive submonoids of Additive M.

Русский

Подмономоды моноида M изоморфны к ADDitive M в виде аддитивных подмономодов.

LaTeX

$$$ Submonoid(M) \cong o AddSubmonoid(Additive M) $$$

Lean4

/-- Submonoids of monoid `M` are isomorphic to additive submonoids of `Additive M`. -/
@[simps]
def toAddSubmonoid : Submonoid M ≃o AddSubmonoid (Additive M)
    where
  toFun
    S :=
    { carrier := Additive.toMul ⁻¹' S
      zero_mem' := S.one_mem'
      add_mem' := fun ha hb => S.mul_mem' ha hb }
  invFun
    S :=
    { carrier := Additive.ofMul ⁻¹' S
      one_mem' := S.zero_mem'
      mul_mem' := fun ha hb => S.add_mem' ha hb }
  left_inv x := by cases x; rfl
  right_inv x := by cases x; rfl
  map_rel_iff' := Iff.rfl

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