Theorems · Theorem · commutative algebra
IsFractionRing.num_den_reduced
∀ (A : Type u_1) [inst : CommRing A] [inst_1 : IsDomain A] [inst_2 : UniqueFactorizationMonoid A] {K : Type u_2}
[inst_3 : Field K] [inst_4 : Algebra A K] [inst_5 : IsFractionRing A K] (x : K),
IsRelPrime (IsFractionRing.num A x) ↑(IsFractionRing.den A x)- Defined in
- Mathlib.RingTheory.Localization.NumDen
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Submonoidstatement · cited by 3,086
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement · cited by 895
- IsFractionRingstatement and proof · cited by 738
- UniqueFactorizationMonoidstatement and proof · cited by 279
- IsRelPrimestatement · cited by 136
- IsFractionRing.numstatement · cited by 22
- IsFractionRing.denstatement · cited by 22
- IsFractionRing.exists_reduced_fractionproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- den_dvd_of_is_rootproof · cited by 2
- OnePoint.exists_mem_SL2proof · cited by 2
- IsFractionRing.associated_den_num_invproof · cited by 1
- IsFractionRing.num_den_uniqueproof · cited by 1
- num_dvd_of_is_rootproof · cited by 1
- IsFractionRing.isUnit_den_iffproof · cited by 0