Theorems · Theorem · commutative algebra

Module.DirectLimit.lift_injective

∀ {R : Type u_1} [inst : Semiring R] {ι : Type u_2} [inst_1 : Preorder ι] {G : ι → Type u_3}
  [inst_2 : (i : ι) → AddCommMonoid (G i)] [inst_3 : (i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j}
  [inst_4 : DecidableEq ι] {P : Type u_4} [inst_5 : AddCommMonoid P] [inst_6 : Module R P] (g : (i : ι) → G i →ₗ[R] P)
  (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) [IsDirectedOrder ι],
  (∀ (i : ι), Function.Injective ⇑(g i)) → Function.Injective ⇑(Module.DirectLimit.lift R ι G f g Hg)

Defined in

Mathlib.Algebra.Colimit.Module

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPreorderAddCommMonoidModuleDecidableEqAddCommMonoidModuleIsDirectedOrder