Theorems · Theorem · category theory

DirectLimit.Algebra.lift_of

∀ {R : Type u_1} {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 : CommSemiring R]
  [inst_5 : (i : ι) → Semiring (G i)] [inst_6 : (i : ι) → Algebra R (G i)]
  [inst_7 : ∀ (i j : ι) (h : i ≤ j), AlgHomClass (T h) R (G i) (G j)] [inst_8 : Nonempty ι] (P : Type u_7)
  [inst_9 : Semiring P] [inst_10 : Algebra R P] (g : (i : ι) → G i →ₐ[R] P)
  (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) (i : ι) (x : G i),
  (DirectLimit.Algebra.lift G f P g Hg) ((DirectLimit.Algebra.of G f i) x) = (g i) x

Defined in

Mathlib.Algebra.Colimit.DirectLimit

Cited by

0 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFunLikeDirectedSystemIsDirectedOrderCommSemiringSemiringAlgebraAlgHomClassNonemptySemiringAlgebra