instPartialOrder — Mathlib

English

If a morphism of sheaves is mono, then every stalk map is mono.

Русский

Если морфизм шефа моно, то каждый stalk-мэп моно.

LaTeX

$$∀ x, Mono((stalkFunctor C x).map f.1)$$

Lean4

instance : PartialOrder (PartialRefinement u s p)
    where
  le v₁ v₂ := v₁.carrier ⊆ v₂.carrier ∧ ∀ i ∈ v₁.carrier, v₁ i = v₂ i
  le_refl _ := ⟨Subset.refl _, fun _ _ => rfl⟩
  le_trans _ _ _ h₁₂ h₂₃ := ⟨Subset.trans h₁₂.1 h₂₃.1, fun i hi => (h₁₂.2 i hi).trans (h₂₃.2 i <| h₁₂.1 hi)⟩
  le_antisymm v₁ v₂ h₁₂
    h₂₁ :=
    have hc : v₁.carrier = v₂.carrier := Subset.antisymm h₁₂.1 h₂₁.1
    PartialRefinement.ext
      (funext fun x =>
        if hx : x ∈ v₁.carrier then h₁₂.2 _ hx else (v₁.apply_eq hx).trans (Eq.symm <| v₂.apply_eq <| hc ▸ hx))
      hc

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