InfiniteAdeleRing.lean`

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11 declarations, 0 contain sorry, 0 declared as axioms

Infinite Adele Ring

Material destined for Mathlib.

Imports (2)

Imported by 6 files

lemma NumberField.InfiniteAdeleRing.mul_apply line 25 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField.InfiniteAdeleRing, NumberField.InfinitePlace, Eq, NumberField.InfinitePlace.Completion, HMul.hMul
Proof uses: rfl
16 instances, coercions and auxiliary constants left out.
instance (instance at line 28) line 28 source
instance (instance at line 32) line 32 source
instance (instance at line 36) line 36 source
Source documentation
The ℝ algebra structure on InfiniteAdeleRing K.
```
noncomputable instance : Algebra ℝ (InfiniteAdeleRing K) :=
…
```
def NumberField.InfiniteAdeleRing.algEquivMixedSpace If K is a number field, this is the ℝ-linear iso between ∏_v|∞ K_v and ℝ^r × ℂ^s, with the usual notation line 41 source
Source documentation
If K is a number field, this is the ℝ-linear iso between ∏_v|∞ K_v and ℝ^r × ℂ^s, with the usual notation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, LinearEquiv, Real, RingHom.id, NumberField.InfiniteAdeleRing, NumberField.mixedEmbedding.mixedSpace, Subtype, NumberField.InfinitePlace, NumberField.InfinitePlace.IsReal, NumberField.InfinitePlace.IsComplex, Complex
Proof uses: LinearEquiv.symm, RingEquiv, RingEquiv.symm, NumberField.InfiniteAdeleRing.ringEquiv_mixedSpace, LinearEquiv.mk, LinearMap.mk, AddHom.mk, Equiv.toFun, RingEquiv.toEquiv, Equiv.invFun
74 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def NumberField.InfiniteAdeleRing.algEquivMixedSpace :
    InfiniteAdeleRing K ≃ₗ[ℝ] (mixedEmbedding.mixedSpace K) := LinearEquiv.symm {
…
```
instance (instance at line 50) line 50 source
instance (instance at line 54) line 54 source
Source comment
should be elsewhere
```
instance : IsModuleTopology ℝ ℂ := .iso Complex.equivRealProdCLM.symm
```
def NumberField.InfiniteAdeleRing.continuousAlgEquivMixedSpace If K is a number field, this is the continuous ℝ-linear iso between ∏_v|∞ K_v and ℝ^r × ℂ^s, with the usual notation line 58 source
Source documentation
If K is a number field, this is the continuous ℝ-linear iso between ∏_v|∞ K_v and ℝ^r × ℂ^s, with the usual notation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ContinuousLinearEquiv, Real, RingHom.id, NumberField.InfiniteAdeleRing, NumberField.mixedEmbedding.mixedSpace, Subtype, NumberField.InfinitePlace, NumberField.InfinitePlace.IsReal, NumberField.InfinitePlace.IsComplex, Complex
Proof uses: LinearEquiv, NumberField.InfiniteAdeleRing.algEquivMixedSpace, ContinuousLinearEquiv.mk
74 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
```
noncomputable def NumberField.InfiniteAdeleRing.continuousAlgEquivMixedSpace :
    InfiniteAdeleRing K ≃L[ℝ] (mixedEmbedding.mixedSpace K)  where
…
```
instance (instance at line 89) line 89 source
def Rat.infinitePlaceCompletionContinuousAlgEquiv The continuous ℤ-algebra isomorphism between Rat.infinitePlace.Completion and ℝ line 95 source
Source documentation
The continuous ℤ-algebra isomorphism between Rat.infinitePlace.Completion and ℝ. (We use continuous ℤ-algebra equivalences in place of continuous ring equivalences since we don't have the latter.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ContinuousAlgEquiv, Int, NumberField.InfinitePlace.Completion, Rat, Rat.infinitePlace, Real, AbsoluteValue, Exists, RingHom, Complex, Eq, NumberField.place
Proof uses: Homeomorph, IsometryEquiv.toHomeomorph, NumberField.InfinitePlace.Completion.isometryEquivRealOfIsReal, Rat.isReal_infinitePlace, RingEquiv, NumberField.InfinitePlace.Completion.ringEquivRealOfIsReal, ContinuousAlgEquiv.mk, AlgEquiv.mk, Homeomorph.toEquiv
72 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
noncomputable def Rat.infinitePlaceCompletionContinuousAlgEquiv :
    Rat.infinitePlace.Completion ≃A[ℤ] ℝ :=
…
```
lemma Rat.infinitePlace_completion_continuousAlgEquiv_apply_algebraMap line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Eq, Real, ContinuousAlgEquiv, Int, NumberField.InfinitePlace.Completion, Rat.infinitePlace, AbsoluteValue, Exists, RingHom, Complex, NumberField.place, Rat.infinitePlaceCompletionContinuousAlgEquiv, Algebra.algebraMap
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, eq_ratCast, map_ratCast, eq_self
86 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.