Theorems · Theorem · order theory
Int.fract_pos
∀ {R : Type u_2} [inst : Ring R] [inst_1 : LinearOrder R] [inst_2 : FloorRing R] {a : R} [IsOrderedRing R],
0 < Int.fract a ↔ a ≠ ↑⌊a⌋The fractional part of a is positive if and only if a ≠ ⌊a⌋.
- Defined in
- Mathlib.Algebra.Order.Floor.Ring
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 37 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.
- LinearOrderstatement and proof · cited by 8,572
- Ringstatement and proof · cited by 7,463
- IsOrderedRingstatement and proof · cited by 777
- FloorRingstatement and proof · cited by 405
- Int.floorstatement · cited by 225
- sub_ne_zeroproof · cited by 119
- Int.fractstatement · cited by 114
- LE.le.lt_iff_neproof · cited by 34
- Int.fract_nonnegproof · cited by 17
Cited by2
Results whose statement or proof uses this declaration.
- Real.exists_rat_eq_convergent'proof · cited by 1
- ContinuousOn.comp_fract'proof · cited by 1