injectiveDimension_lt_iff — Mathlib

English

A collection of basic LT-lemmas under retracts: HasInjectiveDimensionLT respects retracts and related order relations.

Русский

Базовые леммы LT под retract- sinh: LT сохраняется при retract и смежные отношения порядка.

LaTeX

$$$\text{Retract}.\text{injectiveDimensionLT} \text{ preserves LT relations} $$$

Lean4

theorem injectiveDimension_lt_iff {X : C} {n : ℕ} : injectiveDimension X < n ↔ HasInjectiveDimensionLT X n :=
  by
  refine ⟨fun h ↦ ?_, fun h ↦ sInf_lt_iff.2 ?_⟩
  · have : injectiveDimension X ∈ _ := csInf_mem ⟨⊤, by simp⟩
    simp only [Set.mem_setOf_eq] at this
    exact this _ h
  · obtain _ | n := n
    · exact ⟨⊥, fun _ _ ↦ hasInjectiveDimensionLT_of_ge _ 0 _ (by simp), by decide⟩
    ·
      exact
        ⟨n, fun i hi ↦ hasInjectiveDimensionLT_of_ge _ (n + 1) _ (by simpa using hi), by simp [WithBot.lt_add_one_iff]⟩

Похожие утверждения