Theorems · Theorem · group theory
Submonoid.closure_eq_one_union
∀ {M : Type u_1} [inst : MulOneClass M] (s : Set M), ↑(Submonoid.closure s) = {1} ∪ ↑(Subsemigroup.closure s)The Submonoid.closure of a set is the union of {1} and its Subsemigroup.closure.
- Defined in
- Mathlib.Algebra.Group.Submonoid.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MulOneClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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.coestatement and proof · cited by 8,199
- mul_oneproof · cited by 3,885
- Submonoidstatement · cited by 3,086
- one_mulproof · cited by 2,841
- le_antisymmproof · cited by 2,068
- MulOneClassstatement and proof · cited by 1,018
- Subsemigroupstatement · cited by 323
- Submonoid.closurestatement and proof · cited by 167
- OneMemClass.one_memproof · cited by 87
- Submonoid.subset_closureproof · cited by 46
- Subsemigroup.closurestatement and proof · cited by 43
Cited by2
Results whose statement or proof uses this declaration.
- StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_spanproof · cited by 2
- Algebra.adjoin_nonUnitalSubalgebra_eq_spanproof · cited by 0