Theorems · Theorem · functional analysis
StarOrderedRing.lt_iff
∀ {R : Type u_1} [inst : NonUnitalSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [StarOrderedRing R]
{x y : R} [IsCancelAdd R], x < y ↔ ∃ p, p ≠ 0 ∧ p ∈ AddSubmonoid.closure (Set.range fun s => star s * s) ∧ y = x + p- Defined in
- Mathlib.Algebra.Order.Star.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Quot.sound
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Cites11
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
- 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
- IsCancelAddstatement and proof · cited by 79
- lt_iff_le_and_neproof · cited by 47
- StarOrderedRing.le_iffproof · cited by 5
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