Theorems · Theorem · order theory
mul_pos_iff
∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] {a b : R} [ExistsAddOfLE R] [PosMulStrictMono R]
[MulPosStrictMono R] [AddLeftStrictMono R] [AddLeftReflectLT R], 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0- Cited by
- 8 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- mul_posproof · cited by 374
- ExistsAddOfLEstatement and proof · cited by 330
- AddLeftStrictMonostatement and proof · cited by 203
- PosMulStrictMonostatement and proof · cited by 151
- MulPosStrictMonostatement and proof · cited by 94
- mul_pos_of_neg_of_negproof · cited by 40
- AddLeftReflectLTstatement and proof · cited by 33
- pos_and_pos_or_neg_and_neg_of_mul_posproof · cited by 3
Cited by8
Results whose statement or proof uses this declaration.
- Real.one_lt_rpow_iff_of_posproof · cited by 2
- sameRay_smul_smul_of_mul_nonnegproof · cited by 2
- mul_neg_iffproof · cited by 2
- Real.arctan_add_eq_add_piproof · cited by 2
- ContDiffBump.measure_closedBall_div_le_integralproof · cited by 1
- ContDiffBump.normed_le_div_measure_closedBall_rOutproof · cited by 1
- NumberField.house.exists_ne_zero_int_vec_house_leproof · cited by 0
- sub_mul_sub_pos_iffproof · cited by 0