Theorems · Theorem · order theory
Archimedean.arch
∀ {R : Type u_2} {inst : AddCommMonoid R} {inst_1 : PartialOrder R} [self : Archimedean R] (x : R) {y : R},
0 < y → ∃ n, x ≤ n • yFor any two elements x, y such that 0 < y, there exists a natural number n
such that x ≤ n • y.
- Defined in
- Mathlib.Algebra.Order.Archimedean.Defs
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- Archimedean
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddCommMonoidstatement and proof · cited by 12,281
- PartialOrderstatement and proof · cited by 6,410
- Archimedeanstatement and proof · cited by 603
Cited by19
Results whose statement or proof uses this declaration.
- exists_nat_geproof · cited by 20
- exists_lt_nsmulproof · cited by 4
- Filter.Tendsto.atTop_mul_const'proof · cited by 3
- add_one_pow_unbounded_of_posproof · cited by 3
- Filter.Tendsto.atTop_zsmul_constproof · cited by 3
- existsUnique_zsmul_near_of_posproof · cited by 3
- ArchimedeanClass.mk_smulproof · cited by 2
- Filter.Tendsto.const_mul_atTop'proof · cited by 2
- Set.Icc.addNSMul_eq_rightproof · cited by 2
- Archimedean.ratLt_addproof · cited by 1
- Archimedean.ratLt_bddAboveproof · cited by 1
- Archimedean.ratLt_nonemptyproof · cited by 1