Theorems · Theorem · group theory

DirectSum.AddSubmonoidClass.IsHomogeneous.mem_iff

∀ {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) [inst_4 : DirectSum.Decomposition ℳ] {P : Type u_5}
  [inst_5 : SetLike P M] [AddSubmonoidClass P M] (p : P),
  DirectSum.SetLike.IsHomogeneous ℳ p → ∀ {x : M}, x ∈ p ↔ ∀ (i : ι), ↑(((DirectSum.decompose ℳ) x) i) ∈ p

Defined in

Mathlib.Algebra.DirectSum.Decomposition

Cited by

4 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqAddCommMonoidSetLikeAddSubmonoidClassDirectSum.DecompositionSetLikeAddSubmonoidClass