Theorems · Definition · group theory

DirectSum.decomposeAddEquiv

{ι : Type u_1} →
  {M : Type u_3} →
    {σ : Type u_4} →
      [inst : DecidableEq ι] →
        [inst_1 : AddCommMonoid M] →
          [inst_2 : SetLike σ M] →
            [inst_3 : AddSubmonoidClass σ M] →
              (ℳ : ι → σ) → [DirectSum.Decomposition ℳ] → M ≃+ DirectSum ι fun i => ↥(ℳ i)

If M is graded by ι with degree i component ℳ i, then it is isomorphic as an additive monoid to a direct sum of components.

Defined in

Mathlib.Algebra.DirectSum.Decomposition

Cited by

12 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqAddCommMonoidSetLikeAddSubmonoidClassDirectSum.Decomposition