WithZeroMulInt.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

Topological results for integer-valued rings

This file contains topological results for valuation rings taking values in the multiplicative integers with zero adjoined. These are useful for cases where there is a Valued K ℤᵐ⁰ instance but no canonical base with which to embed this into NNReal.

Imports (8)

Imported by 2 files

theorem Valued.WithZeroMulInt.irreducible_valuation_lt_one line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Valued, WithZero, Multiplicative, Int, Subtype, Membership.mem, Subring, Valued.integer, Irreducible, LT.lt, Valuation, Valued.v, OfNat.ofNat
Proof uses: Not, Eq, RingHom, Valuation.integer, Algebra.algebraMap, mt, IsUnit, Iff.mpr, Valuation.Integers.isUnit_iff_valuation_eq_one, Valuation.integer.integers, Irreducible.not_isUnit, lt_of_le_of_ne, Iff.mp, LE.le, Valuation.mem_integer_iff, Subtype.property
80 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Valued.WithZeroMulInt.irreducible_valuation_le_ofAdd_neg_one line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Valued, WithZero, Multiplicative, Int, Subtype, Membership.mem, Subring, Valued.integer, Irreducible, LE.le, Valuation, Valued.v, Equiv, Multiplicative.ofAdd, Neg.neg, OfNat.ofNat
Proof uses: LT.lt, Multiplicative.toAdd, WithZero.unzero, Ne, of_eq_true, Not, Eq, Eq.trans, False, True, congrArg, eq_false, Irreducible.ne_zero, not_false_eq_true, Iff.mp, WithZero.lt_ofAdd_iff, Valued.WithZeroMulInt.irreducible_valuation_lt_one, Eq.mpr, id, propext and 2 more
94 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Valued.WithZeroMulInt.mem_maximalIdeal_pow_valuation line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Valued, WithZero, Multiplicative, Int, IsDiscreteValuationRing, Subtype, Membership.mem, Subring, Valued.integer, Nat, Ideal, HPow.hPow, Valued.maximalIdeal, Irreducible, LE.le, Valuation, Valued.v
Proof uses: dite, Eq, OfNat.ofNat, Classical.propDecidable, of_eq_true, Eq.trans, True, congrFun', congrArg, map_zero, Not, Dvd.dvd, Eq.mp, Ideal.span, Singleton.singleton, Set, Ideal.span_singleton_pow, Irreducible.maximalIdeal_eq, HMul.hMul, Eq.mpr and 10 more
163 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma Valued.WithZeroMulInt.finite_quotient_maximalIdeal_pow_of_finite_residueField Note: this is copied directly from the mathlib result Valued.integer.finite_quotient_maximalIdeal_pow_of_finite_residueField with a relaxation on some of the hypotheses on K line 58 source
Source comment
Note: this is copied directly from the mathlib result `Valued.integer.finite_quotient_maximalIdeal_pow_of_finite_residueField` with a relaxation on some of the hypotheses on `K`. The proof is unchanged.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, LinearOrderedCommGroupWithZero, Valued, IsDiscreteValuationRing, Subtype, Membership.mem, Subring, Valued.integer, Finite, Valued.ResidueField, Nat, HasQuotient.Quotient, Ideal, HPow.hPow, Valued.maximalIdeal
Proof uses: Nat.recAux, Eq.mpr, OfNat.ofNat, Top.top, id, Eq, congrArg, Eq.trans, pow_zero, Ideal.one_eq_top, LE.le, HAdd.hAdd, Ideal.pow_le_pow_right, of_eq_true, True, Ideal.map, RingHom, Ideal.Quotient.mk, Finite.of_equiv, RingEquiv.toEquiv and 9 more
122 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
lemma finite_quotient_maximalIdeal_pow_of_finite_residueField {K Γ₀ : Type*} [Field K]
    [LinearOrderedCommGroupWithZero Γ₀] [Valued K Γ₀] [IsDiscreteValuationRing 𝒪[K]]
    (h : Finite 𝓀[K]) (n : ℕ) :
    Finite (𝒪[K] ⧸ 𝓂[K] ^ n) := by
…
```
theorem Valued.WithZeroMulInt.finite_cover_of_uniformity_basis Documented declaration: finite_cover_of_uniformity_basis line 80 source
Source documentation
The ring of integers 𝒪[K] of a ℤᵐ⁰-valued field K with finite residue field has a finite covering by elements of the basis of uniformity of K, whenever 𝒪[K] is a discrete valuation ring.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Valued, WithZero, Multiplicative, Int, IsDiscreteValuationRing, Subtype, Membership.mem, Subring, Valued.integer, Units, Finite, Valued.ResidueField, Exists, Set, And, Set.Finite, LE.le, Subsemigroup.carrier, Submonoid.toSubsemigroup and 12 more
Proof uses: Irreducible, IsDiscreteValuationRing.exists_irreducible, Nat, HPow.hPow, Valued.exists_pow_lt_of_le_exp_neg_one, Valued.WithZeroMulInt.irreducible_valuation_le_ofAdd_neg_one, HasQuotient.Quotient, Ideal, Valued.maximalIdeal, Valued.WithZeroMulInt.finite_quotient_maximalIdeal_pow_of_finite_residueField, Fintype, Fintype.ofFinite, Set.image, Finset, Finset.image, Quotient, Submodule.quotientRel, Classical.propDecidable, Eq, Quotient.out and 19 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem finite_cover_of_uniformity_basis [IsDiscreteValuationRing 𝒪[K]] (γ : ℤᵐ⁰ˣ)
    (h : Finite 𝓀[K]) :
    ∃ t : Set K, Set.Finite t ∧
      (𝒪[K]).carrier ⊆ ⋃ y ∈ t, { x | (x, y) ∈ { p | v (p.2 - p.1) < γ.val } } := by
…
```
theorem Valued.WithZeroMulInt.integer_compactSpace The ring of integers 𝒪[K] of a complete ℤᵐ⁰-valued field K with finite residue field is compact, whenever 𝒪[K] is a discrete valuation ring line 101 source
Source documentation
The ring of integers 𝒪[K] of a complete ℤᵐ⁰-valued field K with finite residue field is compact, whenever 𝒪[K] is a discrete valuation ring.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Valued, WithZero, Multiplicative, Int, CompleteSpace, IsDiscreteValuationRing, Subtype, Membership.mem, Subring, Valued.integer, Valuation.IsRankOneDiscrete, Valued.v, Finite, Valued.ResidueField, Function.Surjective, Valuation, CompactSpace
Proof uses: CompactSpace.mk, Iff.mp, IsCompact, Set, Set.univ, isCompact_iff_isCompact_univ, Iff.mpr, And, TotallyBounded, IsComplete, isCompact_iff_totallyBounded_isComplete, And.intro, Units, MonoidWithZeroHom.ValueGroup₀, Subgroup, MonoidWithZeroHom.valueGroup, True, Exists, Set.Finite, LE.le and 51 more
131 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem integer_compactSpace [CompleteSpace K] [IsDiscreteValuationRing 𝒪[K]]
    [hv.v.IsRankOneDiscrete] (h : Finite 𝓀[K]) (hsurj : Function.Surjective hv.v) :
    CompactSpace 𝒪[K] where
…
```