Theorems · Theorem · order theory
mul_self_nonneg
∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] [ExistsAddOfLE R] [PosMulMono R] [AddLeftMono R] (a : R),
0 ≤ a * a- Cited by
- 35 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- sqproof · cited by 280
- PosMulMonostatement and proof · cited by 165
- sq_nonnegproof · cited by 106
Cited by35
Results whose statement or proof uses this declaration.
- Even.pow_nonnegproof · cited by 94
- Complex.normSq_nonnegproof · cited by 14
- Real.sqrt_mul_selfproof · cited by 14
- Real.sqrt_le_leftproof · cited by 9
- InnerProductGeometry.cos_angle_add_of_inner_eq_zeroproof · cited by 5
- InnerProductGeometry.sin_angle_add_of_inner_eq_zeroproof · cited by 5
- InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zeroproof · cited by 5
- IsSquare.nonnegproof · cited by 4
- InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zeroproof · cited by 3
- Zsqrtd.norm_nonnegproof · cited by 3
- RootPairing.posRootForm_posForm_pos_of_ne_zeroproof · cited by 3
- InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zeroproof · cited by 3