Theorems · Theorem · category theory

DirectLimit.lift_div

∀ {ι : Type u_2} [inst : Preorder ι] {G : ι → Type u_3} {H : ι → Type u_4} {C : Type u_5}
  {T : ⦃i j : ι⦄ → i ≤ j → Type u_6} {f : (x x_1 : ι) → (h : x ≤ x_1) → T h}
  [inst_1 : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)] [inst_2 : (i : ι) → FunLike (H i) (G i) C]
  [inst_3 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] [inst_4 : IsDirectedOrder ι] [inst_5 : Nonempty ι]
  [inst_6 : (i : ι) → Group (G i)] [inst_7 : Group C]
  [inst_8 : ∀ (i j : ι) (h : i ≤ j), MonoidHomClass (T h) (G i) (G j)] [∀ (i : ι), MonoidHomClass (H i) (G i) C]
  (g : (i : ι) → H i) (h : ∀ (i j : ι) (h : i ≤ j) (x : G i), (g i) x = (g j) ((f i j h) x)) (x y : DirectLimit G f),
  DirectLimit.lift f (fun x => ⇑(g x)) h (x / y) =
    DirectLimit.lift f (fun x => ⇑(g x)) h x / DirectLimit.lift f (fun x => ⇑(g x)) h y

Defined in

Mathlib.Algebra.Colimit.DirectLimit

Cited by

0 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFunLikeFunLikeDirectedSystemIsDirectedOrderNonemptyGroupGroupMonoidHomClassMonoidHomClass