Theorems · Theorem · group theory
addIrreducible_mem_addSubmonoidClosure_subset
∀ {M : Type u_1} [inst : AddCommMonoid M] [Subsingleton (AddUnits M)] {S : Set M},
{p | p ∈ AddSubmonoid.closure S ∧ AddIrreducible p} ⊆ SAny set S inside an additive monoid with a single unit contains the irreducible
elements of the submonoid it generates.
- Defined in
- Mathlib.Algebra.AffineMonoid.Irreducible
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoidSubsingleton
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- AddCommMonoidstatement and proof · cited by 12,281
- Set.ofPredstatement and proof · cited by 6,101
- add_zeroproof · cited by 2,707
- zero_addproof · cited by 2,366
- AddSubmonoidstatement · cited by 1,178
- AddUnitsstatement and proof · cited by 325
- AddSubmonoid.closurestatement and proof · cited by 224
- IsEmpty.forall_iffproof · cited by 39
- AddSubmonoid.closure_inductionproof · cited by 30
- AddIrreduciblestatement and proof · cited by 24
- not_addIrreducible_zeroproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- addIrreducible_subset_of_addSubmonoidClosure_eq_topproof · cited by 1
- AddSubmonoid.FG.finite_addIrreducible_mem_addSubmonoidClosureproof · cited by 1