Project-declaredLean 4.32.0 · mathlib@249c48c2

Number Field Adele Ring units mem ring Haar Character ker

NumberField.AdeleRing.units_mem_ringHaarCharacter_ker

number theory arithmetic geometry Fermat's Last Theorem adele ring

Plain-language statement

Left multiplication by an element of Bˣ on B ⊗ 𝔸_K does not scale additive Haar measure. In other words, Bˣ is in the kernel of the ringHaarChar of B ⊗ 𝔸_K.

Exact Lean statement

lemma NumberField.AdeleRing.units_mem_ringHaarCharacter_ker
    [MeasurableSpace (B ⊗[K] AdeleRing (𝓞 K) K)] [BorelSpace (B ⊗[K] AdeleRing (𝓞 K) K)]
    (b : Bˣ) :
    (Units.map Algebra.TensorProduct.includeLeftRingHom.toMonoidHom b :
      (B ⊗[K] AdeleRing (𝓞 K) K)ˣ) ∈
    ringHaarCharKer (B ⊗[K] AdeleRing (𝓞 K) K)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma NumberField.AdeleRing.units_mem_ringHaarCharacter_ker    [MeasurableSpace (B ⊗[K] AdeleRing (𝓞 K) K)] [BorelSpace (B ⊗[K] AdeleRing (𝓞 K) K)]    (b : Bˣ) :    (Units.map Algebra.TensorProduct.includeLeftRingHom.toMonoidHom b :      (B ⊗[K] AdeleRing (𝓞 K) K)ˣ) ∈    ringHaarCharKer (B ⊗[K] AdeleRing (𝓞 K) K) := by  rw [mem_ringHaarChar_ker, ringHaarChar_apply]  convert! MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one K B (LinearEquiv.mulLeft K b)  ext c  change _ = (ContinuousLinearEquiv.baseChange K _ _ _ _) c  induction c with  | zero => simp  | tmul x y => simp [LinearEquiv.mulLeft]  | add x y hx hy => simp_all [mul_add]

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/HaarMeasure/HaarChar/AdeleRing.lean:222-235

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