Theorems · Theorem · order theory
exists_nat_pow_near
∀ {R : Type u_3} [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] [Archimedean R] [ExistsAddOfLE R]
{x y : R}, 1 ≤ x → 1 < y → ∃ n, y ^ n ≤ x ∧ x < y ^ (n + 1)Every x greater than or equal to 1 is between two successive natural-number powers of every y greater than one.
- Defined in
- Mathlib.Algebra.Order.Archimedean.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- LinearOrderstatement and proof · cited by 8,572
- IsStrictOrderedRingstatement and proof · cited by 2,490
- pow_zeroproof · cited by 1,094
- ne_of_gtproof · cited by 637
- Archimedeanstatement and proof · cited by 603
- le_of_not_gtproof · cited by 430
- ExistsAddOfLEstatement and proof · cited by 330
- pos_iff_ne_zeroproof · cited by 180
- Nat.findproof · cited by 139
- not_le_of_gtproof · cited by 97
- Nat.find_specproof · cited by 74
Cited by2
Results whose statement or proof uses this declaration.
- exists_nat_pow_near_of_lt_oneproof · cited by 2
- Sylow.exists_subgroup_le_card_leproof · cited by 1