instMonoid — Mathlib

English

An L-equivalence f:M ≃[L] N has range equal to the top substructure, i.e., its underlying homomorphism is surjective.

Русский

Тождественная эквивалентность L между M и N имеет образ, равный верхней подструктуре, то есть гомоморфизм на самом деле сюръективен.

LaTeX

$$$f.toHom.range = \\top$$$

Lean4

instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      intro ⟨y₁, y₂⟩ ymem ynotMem
      have y2ne : y₂ ≠ 1 := by
        intro con
        simp_all
      simp [y2ne]
    mul_assoc := mul_smul' }

Похожие утверждения