Theorems · Theorem · ring theory

CoalgHom.mk_coe

∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A]
  [inst_3 : AddCommMonoid B] [inst_4 : Module R B] [inst_5 : CoalgebraStruct R A] [inst_6 : CoalgebraStruct R B]
  {f : A →ₗc[R] B} (h₁ : ∀ (x y : A), f (x + y) = f x + f y)
  (h₂ :
    ∀ (m : R) (x : A),
      { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₁ }.toFun x)
  (h₃ : CoalgebraStruct.counit ∘ₗ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } = CoalgebraStruct.counit)
  (h₄ :
    TensorProduct.map { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }
          { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } ∘ₗ
        CoalgebraStruct.comul =
      CoalgebraStruct.comul ∘ₗ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }),
  { toFun := ⇑f, map_add' := h₁, map_smul' := h₂, counit_comp := h₃, map_comp_comul := h₄ } = f

Defined in

Mathlib.RingTheory.Coalgebra.Hom

Cited by

0 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleCoalgebraStructCoalgebraStruct