Theorems · Theorem · group theory
IsOfFinOrder.mem_powers_iff_mem_range_orderOf
∀ {G : Type u_1} [inst : Monoid G] {x y : G} [inst_1 : DecidableEq G],
IsOfFinOrder x → (y ∈ Submonoid.powers x ↔ y ∈ Finset.image (fun x_1 => x ^ x_1) (Finset.range (orderOf x)))- Defined in
- Mathlib.GroupTheory.OrderOfElement
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MonoidDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement · cited by 13,712
- Monoidstatement and proof · cited by 3,887
- Submonoidstatement · cited by 3,086
- Finset.rangestatement · cited by 1,341
- Finset.imagestatement · cited by 910
- Submonoid.powersstatement · cited by 408
- orderOfstatement · cited by 324
- IsOfFinOrderstatement and proof · cited by 113
- IsOfFinOrder.orderOf_posproof · cited by 17
- pow_mod_orderOfproof · cited by 12
- Finset.mem_range_iff_mem_finset_range_of_mod_eq'proof · cited by 4
Cited by2
Results whose statement or proof uses this declaration.
- IsOfFinOrder.powers_eq_image_range_orderOfproof · cited by 2
- IsOfFinOrder.mem_zpowers_iff_mem_range_orderOfproof · cited by 1