Theorems · Theorem · category theory

DirectLimit.mul_def

∀ {ι : Type u_2} [inst : Preorder ι] {G : ι → Type u_3} {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 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] [inst_3 : IsDirectedOrder ι] [inst_4 : (i : ι) → Mul (G i)]
  [inst_5 : ∀ (i j : ι) (h : i ≤ j), MulHomClass (T h) (G i) (G j)] (i : ι) (x y : G i),
  ⟦⟨i, x⟩⟧ * ⟦⟨i, y⟩⟧ = ⟦⟨i, x * y⟩⟧

Defined in

Mathlib.Algebra.Colimit.DirectLimit

Cited by

2 results in Mathlib

Foundations

Depth 18 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFunLikeDirectedSystemIsDirectedOrderMulMulHomClass