IsDiscreteValuationRing.lean`

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1 declarations, 0 contain sorry, 0 declared as axioms

Elements of valuation one

Material for Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing: an element of the fraction field of a discrete valuation ring with valuation 1 is the image of a unit.

Imports (2)

Imported by 2 files

theorem exists_algebraMap_unit_eq_of_valuation_eq_one An element of the fraction field of a discrete valuation ring with valuation 1 is the image of a unit line 30 source
Source documentation
An element of the fraction field of a discrete valuation ring with valuation 1 is the image of a unit.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDomain, IsDiscreteValuationRing, Field, Algebra, IsFractionRing, Eq, WithZero, Multiplicative, Int, Valuation, IsDedekindDomain.HeightOneSpectrum.valuation, IsDiscreteValuationRing.maximalIdeal, OfNat.ofNat, Exists, Units, RingHom, Algebra.algebraMap
Proof uses: HSMul.hSMul, IsDiscreteValuationRing.associated_of_valuation_eq, Eq.mpr, id, congrArg, map_one, HMul.hMul, Eq.symm, Algebra.smul_def, Inv.inv, map_mul, Units.val_mul, mul_inv_cancel, Units.val_one, Eq.refl, Exists.intro, mul_left_cancel₀, left_ne_zero_of_mul_eq_one, Eq.trans
111 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem exists_algebraMap_unit_eq_of_valuation_eq_one {x : K}
    (hx : valuation K (maximalIdeal R) x = 1) : ∃ u : Rˣ, algebraMap R K ↑u = x := by
…
```