Theorems · Theorem · functional analysis
LE.le.star_eq
∀ {R : Type u_1} [inst : NonUnitalSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [StarOrderedRing R]
{x : R}, 0 ≤ x → star x = xThe combination (IsSelfAdjoint.star_eq <| .of_nonneg ·) for use with dot notation.
- Defined in
- Mathlib.Algebra.Order.Star.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, 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 · cited by 1,082
- StarOrderedRingstatement and proof · cited by 587
- NonUnitalSemiringstatement and proof · cited by 339
- IsSelfAdjoint.star_eqproof · cited by 58
- LE.le.isSelfAdjointproof · cited by 20
Cited by8
Results whose statement or proof uses this declaration.
- CFC.norm_star_mul_mul_self_of_nonnegproof · cited by 3
- CFC.abs_of_nonnegproof · cited by 2
- CFC.norm_mul_mul_star_self_of_nonnegproof · cited by 1
- CStarAlgebra.nnnorm_sub_mul_self_leproof · cited by 1
- IsStarProjection.mul_right_and_mul_left_of_nonneg_of_leproof · cited by 1
- PositiveLinearMap.preGNS_norm_sqproof · cited by 0
- CFC.norm_absproof · cited by 0