Theorems · Definition · number theory
PadicInt.withValIntegersRingEquiv
{p : ℕ} → [inst : Fact (Nat.Prime p)] → ↥(Valued.integer (Rat.padicValuation p).Completion) ≃+* ℤ_[p]The p-adic integers are ring isomorphic to the integers of the uniform completion
of the rationals at the p-adic valuation.
- Defined in
- Mathlib.NumberTheory.Padics.WithVal
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Fact
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Factstatement and proof · cited by 2,726
- Nat.Primestatement and proof · cited by 2,059
- RingEquivstatement · cited by 1,147
- Multiplicativestatement · cited by 875
- Subringstatement · cited by 602
- WithZerostatement · cited by 586
- PadicIntstatement · cited by 179
- WithValstatement · cited by 151
- Valuation.Completionstatement and proof · cited by 33
- Valued.integerstatement and proof · cited by 22
- Rat.padicValuationstatement and proof · cited by 18
- PadicInt.subringproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Rat.HeightOneSpectrum.adicCompletionIntegers.padicIntEquivproof · cited by 4