Theorems · Theorem · group theory
AddCommute.exists_addOrderOf_eq_lcm
∀ (G : Type u) [inst : AddMonoid G] {x y : G},
AddCommute x y → ∃ z ∈ AddSubmonoid.closure {x, y}, addOrderOf z = (addOrderOf x).lcm (addOrderOf y)If two commuting elements x and y of an additive monoid have order n and
m, then there is an element of order lcm n m that lies in the additive subgroup generated by x
and y.
- Defined in
- Mathlib.GroupTheory.Exponent
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddMonoid
Around this declaration
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Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- AddMonoidstatement and proof · cited by 2,864
- AddSubmonoidstatement · cited by 1,178
- AddMemClass.add_memproof · cited by 229
- AddSubmonoid.closurestatement · cited by 224
- addOrderOfstatement and proof · cited by 208
- AddCommutestatement and proof · cited by 185
- Set.mem_singleton_iffproof · cited by 172
- AddSubmonoid.subset_closureproof · cited by 63
- Set.mem_insert_iffproof · cited by 39
- nsmul_memproof · cited by 26
- Nat.factorizationLCMLeftproof · cited by 15
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