Theorems · Theorem · group theory
irreducible_mem_submonoidClosure_subset
∀ {M : Type u_1} [inst : CommMonoid M] [Subsingleton Mˣ] {S : Set M}, {p | p ∈ Submonoid.closure S ∧ Irreducible p} ⊆ SAny set S inside a 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
- CommMonoidSubsingleton
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Set.ofPredstatement and proof · cited by 6,101
- mul_oneproof · cited by 3,885
- Submonoidstatement · cited by 3,086
- one_mulproof · cited by 2,841
- Unitsstatement and proof · cited by 2,804
- CommMonoidstatement and proof · cited by 2,264
- Irreduciblestatement and proof · cited by 496
- Submonoid.closurestatement and proof · cited by 167
- Submonoid.closure_inductionproof · cited by 27
- Irreducible.eq_one_or_eq_oneproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Submonoid.FG.finite_irreducible_mem_submonoidClosureproof · cited by 1
- irreducible_subset_of_submonoidClosure_eq_topproof · cited by 1