Theorems · Theorem · group theory
Subsemigroup.isMulCommutative_iSup
∀ {ι : Sort u_1} {M : Type u_2} [inst : Mul M] {S : ι → Subsemigroup M} [hS : ∀ (i : ι), IsMulCommutative ↥(S i)],
Directed (fun x1 x2 => x1 ≤ x2) S → IsMulCommutative ↥(⨆ i, S i)The supremum of a directed family of commutative subsemigroups is commutative.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MulIsMulCommutative
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- iSupstatement · cited by 2,415
- Subsemigroupstatement and proof · cited by 323
- Directedstatement and proof · cited by 213
- IsMulCommutativestatement and proof · cited by 95
- IsMulCommutative.of_setLike_mul_commproof · cited by 16
- setLike_mul_commproof · cited by 6
- Subsemigroup.coe_iSup_of_directedproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- Subsemiring.isMulCommutative_iSupproof · cited by 1
- Subring.isMulCommutative_iSupproof · cited by 0