Theorems · Theorem · category theory

DirectLimit.NonUnitalAlgebra.hom_ext_iff

∀ {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) → NonUnitalNonAssocSemiring (G i)] [inst_6 : (i : ι) → DistribMulAction R (G i)]
  [inst_7 : ∀ (i j : ι) (h : i ≤ j), NonUnitalAlgHomClass (T h) R (G i) (G j)] [inst_8 : Nonempty ι] {P : Type u_7}
  [inst_9 : NonUnitalNonAssocSemiring P] [inst_10 : DistribMulAction R P] {g₁ g₂ : DirectLimit G f →ₙₐ[R] P},
  g₁ = g₂ ↔ ∀ (i : ι), g₁.comp (DirectLimit.NonUnitalAlgebra.of G f i) = g₂.comp (DirectLimit.NonUnitalAlgebra.of G f i)

Defined in

Mathlib.Algebra.Colimit.DirectLimit

Cited by

0 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFunLikeDirectedSystemIsDirectedOrderCommSemiringNonUnitalNonAssocSemiringDistribMulActionNonUnitalAlgHomClassNonemptyNonUnitalNonAssocSemiringDistribMulAction