Theorems · Theorem · functional analysis
star_mul_self_pos
∀ {R : Type u_1} [inst : NonUnitalSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [StarOrderedRing R]
[Nontrivial R] {x : R}, IsRegular x → 0 < star x * x- Defined in
- Mathlib.Algebra.Order.Star.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- Nontrivialstatement and proof · cited by 2,416
- MulZeroClass.mul_zeroproof · cited by 2,091
- 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
- IsRegularstatement and proof · cited by 116
- IsRegular.leftproof · cited by 39
- LE.le.lt_iff_neproof · cited by 34
- star_mul_self_nonnegproof · cited by 17
- IsRegular.ne_zeroproof · cited by 7
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.dotProduct_star_self_pos_iffproof · cited by 1
- mul_star_self_posproof · cited by 0