Theorems · Theorem · group theory
Submonoid.mem_closure_singleton
∀ {M : Type u_1} [inst : Monoid M] {x y : M}, y ∈ Submonoid.closure {x} ↔ ∃ n, x ^ n = yThe submonoid generated by an element of a monoid equals the set of natural number powers of the element.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Monoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Monoidstatement and proof · cited by 3,887
- Submonoidstatement and proof · cited by 3,086
- Submonoid.closurestatement · cited by 167
- MonoidHom.mem_mrangeproof · cited by 2
- Submonoid.closure_singleton_eqproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Submonoid.powers_eq_closureproof · cited by 6
- Submonoid.mem_closure_singleton_selfproof · cited by 0