Theorems · Theorem · order theory
one_lt_div
∀ {α : Type u_2} [inst : Semifield α] [inst_1 : PartialOrder α] [PosMulReflectLT α] {a b : α},
0 < b → (1 < a / b ↔ b < a)- Defined in
- Mathlib.Algebra.Order.Field.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- PartialOrderstatement and proof · cited by 6,410
- one_mulproof · cited by 2,841
- Semifieldstatement and proof · cited by 439
- PosMulReflectLTstatement and proof · cited by 278
- lt_div_iff₀proof · cited by 44
Cited by12
Results whose statement or proof uses this declaration.
- ContDiffBump.one_lt_rOut_div_rInproof · cited by 3
- NNReal.strictConcaveOn_rpowproof · cited by 3
- isLittleO_pow_const_mul_const_pow_const_pow_of_norm_ltproof · cited by 2
- strictConvexOn_rpowproof · cited by 2
- Rat.den_le_and_le_num_le_of_sub_lt_one_div_den_sqproof · cited by 1
- Real.exists_natCast_add_one_lt_pow_of_one_ltproof · cited by 1
- le_iff_forall_one_lt_le_mul₀proof · cited by 1
- MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_innerproof · cited by 1
- AbsoluteValue.IsEquiv.log_div_log_eq_log_div_logproof · cited by 1
- Bound.one_lt_div_of_pos_of_ltproof · cited by 0
- AnalyticOnNhd.sum_divisor_leproof · cited by 0