Theorems · Theorem · order theory
mul_nonneg_iff
∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] {a b : R} [ExistsAddOfLE R] [MulPosStrictMono R]
[PosMulStrictMono R] [AddLeftReflectLE R] [AddLeftMono R], 0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0- Cited by
- 6 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
- AddLeftMonostatement and proof · cited by 687
- mul_nonnegproof · cited by 397
- ExistsAddOfLEstatement and proof · cited by 330
- PosMulStrictMonostatement and proof · cited by 151
- AddLeftReflectLEstatement and proof · cited by 119
- MulPosStrictMonostatement and proof · cited by 94
- mul_nonneg_of_nonpos_of_nonposproof · cited by 45
- nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nonnegproof · cited by 1
Cited by6
Results whose statement or proof uses this declaration.
- mul_nonneg_iff_pos_imp_nonnegproof · cited by 3
- mul_nonpos_iffproof · cited by 2
- Int.nneg_mul_add_sq_of_abs_le_oneproof · cited by 1
- Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁'proof · cited by 1
- mul_nonneg_of_threeproof · cited by 0
- sub_mul_sub_nonneg_iffproof · cited by 0