Theorems · Theorem · order theory
mul_nonneg_iff_pos_imp_nonneg
∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] {a b : R} [ExistsAddOfLE R] [PosMulStrictMono R]
[MulPosStrictMono R] [AddLeftMono R] [AddLeftReflectLE R], 0 ≤ a * b ↔ (0 < a → 0 ≤ b) ∧ (0 < b → 0 ≤ a)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- AddLeftMonostatement and proof · cited by 687
- ExistsAddOfLEstatement and proof · cited by 330
- le_totalproof · cited by 294
- PosMulStrictMonostatement and proof · cited by 151
- AddLeftReflectLEstatement and proof · cited by 119
- MulPosStrictMonostatement and proof · cited by 94
- mul_nonneg_iffproof · cited by 6
Cited by3
Results whose statement or proof uses this declaration.
- mul_nonneg_iff_neg_imp_nonposproof · cited by 0
- mul_nonpos_iff_pos_imp_nonposproof · cited by 0
- mul_nonpos_iff_neg_imp_nonnegproof · cited by 0