Theorems · Theorem · order theory
AddCommGroup.modEq_iff_forall_notMem_Ioo_mod
∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : LinearOrder α] [IsOrderedAddMonoid α] [hα : Archimedean α] {p : α},
0 < p → ∀ {a b : α}, a ≡ b [PMOD p] ↔ ∀ (z : ℤ), b - z • p ∉ Set.Ioo a (a + p)- Defined in
- Mathlib.Algebra.Order.ToIntervalMod
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- Setstatement · cited by 53,352
- AddCommGroupstatement and proof · cited by 12,871
- LinearOrderstatement and proof · cited by 8,572
- IsOrderedAddMonoidstatement and proof · cited by 1,659
- Set.Ioostatement and proof · cited by 1,214
- Archimedeanstatement and proof · cited by 603
- List.TFAE.outproof · cited by 177
- toIocModproof · cited by 109
- AddCommGroup.ModEqstatement and proof · cited by 107
- toIcoModproof · cited by 99
- AddCommGroup.tfae_modEqproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Ico_eq_locus_Ioc_eq_iUnion_Iooproof · cited by 1