Theorems · Theorem · ring theory

DirectSum.toModule.unique

∀ (R : Type u) [inst : Semiring R] {ι : Type v} {M : ι → Type w} [inst_1 : (i : ι) → AddCommMonoid (M i)]
  [inst_2 : (i : ι) → Module R (M i)] [inst_3 : DecidableEq ι] {N : Type u₁} [inst_4 : AddCommMonoid N]
  [inst_5 : Module R N] (ψ : (DirectSum ι fun i => M i) →ₗ[R] N) (f : DirectSum ι fun i => M i),
  ψ f = (DirectSum.toModule R ι N fun i => ψ ∘ₗ DirectSum.lof R ι M i) f

Every linear map from a direct sum agrees with the one obtained by applying the universal property to each of its components.

Defined in

Mathlib.Algebra.DirectSum.Module

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleDecidableEqAddCommMonoidModule