Theorems · Theorem · number theory

NumberField.HeightOneSpectrum.adicAbv_add_le_max

∀ (K : Type u_1) [inst : Field K] {R : Type u_2} [inst_1 : CommRing R] [inst_2 : Algebra R K]
  [inst_3 : IsDedekindDomain R] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
  [inst_5 : Module.Finite ℤ R] [inst_6 : Module.Free ℤ R] (x y : K),
  (NumberField.HeightOneSpectrum.adicAbv K v) (x + y) ≤
    max ((NumberField.HeightOneSpectrum.adicAbv K v) x) ((NumberField.HeightOneSpectrum.adicAbv K v) y)

The v-adic absolute value satisfies the ultrametric inequality.

Defined in: Mathlib.NumberTheory.NumberField.Completion.FinitePlace
Cited by: 3 results in Mathlib
Foundations: Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field CommRing Algebra IsDedekindDomain IsFractionRing Module.Finite Module.Free

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Cites13

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement · cited by 62,936
Realstatement · cited by 25,697
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
Fieldstatement and proof · cited by 7,404
Module.Finitestatement and proof · cited by 1,032
IsFractionRingstatement and proof · cited by 738
IsDedekindDomainstatement and proof · cited by 668
Module.Freestatement and proof · cited by 597
AbsoluteValuestatement · cited by 363
IsDedekindDomain.HeightOneSpectrumstatement and proof · cited by 338
NumberField.HeightOneSpectrum.adicAbvstatement · cited by 21

Cited by3

Results whose statement or proof uses this declaration.

NumberField.FinitePlace.add_leproof · cited by 1
NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_add_le_maxproof · cited by 0
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_add_le_maxproof · cited by 0