instGroupWithZero — Mathlib

English

Let α be nonempty and b ∈ range D with hmem: D.invFun b ∈ s and hmin: ∀ a ∈ s, b ≤ D(a). Then s.inf D = b.

Русский

Пусть α непусто и b ∈ range(D) с hmem: D^{-1}(b) ∈ s и hmin: ∀ a ∈ s, b ≤ D(a). Тогда s.inf D = b.

LaTeX

$$$b \\in \\mathrm{range}(D) \\land D^{-1}(b) \\in s \\land \\forall a \\in s, b \\le D(a) \\Rightarrow s.inf D = b$$$

Lean4

instance instGroupWithZero [GroupWithZero β] : GroupWithZero β*
    where
  __ := instDivInvMonoid
  __ := instMonoidWithZero
  mul_inv_cancel
    f :=
    inductionOn f fun f hf ↦
      coe_eq.2 <| (φ.em fun y ↦ f y = 0).elim (fun H ↦ (hf <| coe_eq.2 H).elim) fun H ↦ H.mono fun _ ↦ mul_inv_cancel₀
  inv_zero := coe_eq.2 <| by simp only [Function.comp_def, inv_zero, EventuallyEq.rfl]

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