Theorems · Theorem · functional analysis
star_mul_self_nonneg
∀ {R : Type u_1} [inst : NonUnitalSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [StarOrderedRing R]
(r : R), 0 ≤ star r * r- Defined in
- Mathlib.Algebra.Order.Star.Basic
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext, Quot.sound
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.
- PartialOrderstatement and proof · cited by 6,410
- StarRingstatement and proof · cited by 1,686
- Star.starstatement and proof · cited by 1,082
- StarOrderedRingstatement and proof · cited by 587
- NonUnitalSemiringstatement and proof · cited by 339
- AddSubmonoid.subset_closureproof · cited by 63
- StarOrderedRing.nonneg_iffproof · cited by 2
Cited by17
Results whose statement or proof uses this declaration.
- star_left_conjugate_nonnegproof · cited by 6
- CFC.abs_mul_absproof · cited by 6
- mul_star_self_nonnegproof · cited by 5
- nonneg_iff_isSelfAdjoint_and_quasispectrumRestrictsproof · cited by 4
- Matrix.posSemidef_conjTranspose_mul_selfproof · cited by 3
- IsSelfAdjoint.mul_self_nonnegproof · cited by 3
- dotProduct_star_self_eq_zeroproof · cited by 2
- star_mul_self_posproof · cited by 2
- CFC.abs_smul_nonnegproof · cited by 2
- Matrix.dotProduct_star_self_pos_iffproof · cited by 1
- Commute.cfcAbs_mul_eqproof · cited by 1