Theorems · Theorem · functional analysis
StarOrderedRing.nonneg_iff
∀ {R : Type u_1} [inst : NonUnitalSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [StarOrderedRing R]
{x : R}, 0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s => star s * s)- Defined in
- Mathlib.Algebra.Order.Star.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext, 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.
- PartialOrderstatement and proof · cited by 6,410
- Set.rangestatement and proof · cited by 4,705
- zero_addproof · cited by 2,366
- StarRingstatement and proof · cited by 1,686
- AddSubmonoidstatement · cited by 1,178
- Star.starstatement and proof · cited by 1,082
- StarOrderedRingstatement and proof · cited by 587
- NonUnitalSemiringstatement and proof · cited by 339
- AddSubmonoid.closurestatement and proof · cited by 224
Cited by2
Results whose statement or proof uses this declaration.
- star_mul_self_nonnegproof · cited by 17
- star_left_conjugate_nonnegproof · cited by 6