Theorems · Theorem · ring theory
NonUnitalSubring.isMulCommutative_closure
∀ {R : Type u_1} [inst : NonUnitalRing R] {s : Set R},
(∀ x ∈ s, ∀ y ∈ s, x * y = y * x) → IsMulCommutative ↥(NonUnitalSubring.closure s)If all the elements of a set s commute, then closure s is a non-unital commutative
semiring.
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NonUnitalRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- SetLike.coeproof · cited by 8,199
- NonUnitalRingstatement and proof · cited by 422
- NonUnitalSubringstatement · cited by 185
- IsMulCommutativestatement · cited by 95
- NonUnitalSubring.closurestatement and proof · cited by 23
- IsMulCommutative.of_setLike_mul_commproof · cited by 16
- Set.centralizer_centralizer_comm_of_commproof · cited by 11
- NonUnitalSubring.centralizerproof · cited by 10
- NonUnitalSubring.closure_le_centralizer_centralizerproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- NonUnitalSubring.closureNonUnitalCommRingOfCommproof · cited by 0