-- Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h₁ : MeasurableSet (α := mixedSpace K) (normAtAllPlaces ⁻¹' compactSet K)
h₂ :
  MeasurableSet (α := mixedSpace K)
    (normAtAllPlaces ⁻¹' (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K))
rw :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in normAtAllPlaces '' normAtAllPlaces ⁻¹' compactSet K,
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K)) =
    ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞)
      (normAtAllPlaces ⁻¹' (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K))
rw₁ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in compactSet K,
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K)) =
    ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞)
      (normAtAllPlaces ⁻¹' (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K))
rw₂ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in compactSet K,
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K)) =
    HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in
        normAtAllPlaces '' normAtAllPlaces ⁻¹' (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K),
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K))
rw₃ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in compactSet K,
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K)) =
    HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K),
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K))
rw₄ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in (↑expMapBasis : realSpace K → realSpace K) '' closure (paramSet K),
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K)) =
    HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K),
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K))
rw₅ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * ENNReal.ofReal (regulator K) * (↑(finrank ℚ K) : ℝ≥0∞) *
        ∫⁻ (x : realSpace K) in closure (paramSet K),
          ENNReal.ofReal (Real.exp (x w₀ * (↑(finrank ℚ K) : ℝ))) *
              (∏ i : { w : InfinitePlace K // w.IsComplex },
                  ENNReal.ofReal
                    ((↑expMapBasis : realSpace K → realSpace K) (fun (w : InfinitePlace K) ↦ x w)
                      (↑i : InfinitePlace K)))⁻¹ *
            ∏ w : { w : InfinitePlace K // w.IsComplex },
              ENNReal.ofReal ((↑expMapBasis : realSpace K → realSpace K) x (↑w : InfinitePlace K))) =
    HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K),
        ∏ w : { w : InfinitePlace K // w.IsComplex }, ENNReal.ofReal (x (↑w : InfinitePlace K))
rw₆ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * ENNReal.ofReal (regulator K) * (↑(finrank ℚ K) : ℝ≥0∞) *
        ∫⁻ (x : realSpace K) in closure (paramSet K),
          ENNReal.ofReal (Real.exp (x w₀ * (↑(finrank ℚ K) : ℝ))) *
              (∏ i : { w : InfinitePlace K // w.IsComplex },
                  ENNReal.ofReal
                    ((↑expMapBasis : realSpace K → realSpace K) (fun (w : InfinitePlace K) ↦ x w)
                      (↑i : InfinitePlace K)))⁻¹ *
            ∏ w : { w : InfinitePlace K // w.IsComplex },
              ENNReal.ofReal ((↑expMapBasis : realSpace K → realSpace K) x (↑w : InfinitePlace K))) =
    HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * ENNReal.ofReal (regulator K) * (↑(finrank ℚ K) : ℝ≥0∞) *
        ∫⁻ (x : realSpace K) in interior (paramSet K),
          ENNReal.ofReal (Real.exp (x w₀ * (↑(finrank ℚ K) : ℝ))) *
              (∏ i : { w : InfinitePlace K // w.IsComplex },
                  ENNReal.ofReal
                    ((↑expMapBasis : realSpace K → realSpace K) (fun (w : InfinitePlace K) ↦ x w)
                      (↑i : InfinitePlace K)))⁻¹ *
            ∏ w : { w : InfinitePlace K // w.IsComplex },
              ENNReal.ofReal ((↑expMapBasis : realSpace K → realSpace K) x (↑w : InfinitePlace K)))
rw₇ rw₈ :
  HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * ENNReal.ofReal (regulator K) * (↑(finrank ℚ K) : ℝ≥0∞) *
        ∫⁻ (x : realSpace K) in interior (paramSet K),
          ENNReal.ofReal (Real.exp (x w₀ * (↑(finrank ℚ K) : ℝ))) *
              (∏ i : { w : InfinitePlace K // w.IsComplex },
                  ENNReal.ofReal
                    ((↑expMapBasis : realSpace K → realSpace K) (fun (w : InfinitePlace K) ↦ x w)
                      (↑i : InfinitePlace K)))⁻¹ *
            ∏ w : { w : InfinitePlace K // w.IsComplex },
              ENNReal.ofReal ((↑expMapBasis : realSpace K → realSpace K) x (↑w : InfinitePlace K))) =
    HPow.hPow (α := ℝ≥0∞) 2 (nrRealPlaces K) * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * ENNReal.ofReal (regulator K) * (↑(finrank ℚ K) : ℝ≥0∞) *
        ∫⁻ (x : realSpace K) in interior (paramSet K),
          ENNReal.ofReal (Real.exp (x w₀ * (↑(finrank ℚ K) : ℝ))) *
              (∏ i : { w : InfinitePlace K // w.IsComplex },
                  ENNReal.ofReal
                    ((↑expMapBasis : realSpace K → realSpace K) (fun (w : InfinitePlace K) ↦ x w)
                      (↑i : InfinitePlace K)))⁻¹ *
            ∏ w : { w : InfinitePlace K // w.IsComplex },
              ENNReal.ofReal ((↑expMapBasis : realSpace K → realSpace K) x (↑w : InfinitePlace K)))
⊢ Eq (α := ℝ≥0∞)
    (((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞) (normAtAllPlaces ⁻¹' compactSet K) : ℝ≥0∞)
    (((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞)
        (normAtAllPlaces ⁻¹' (↑expMapBasis : realSpace K → realSpace K) '' interior (paramSet K)) :
      ℝ≥0∞)

Tactic:
rw [volume_eq_two_pow_mul_two_pi_pow_mul_integral Set.preimage_image_preimage h₁,
    normAtAllPlaces_image_preimage_of_nonneg (fun x a w ↦ nonneg_of_mem_compactSet K a w),
    volume_eq_two_pow_mul_two_pi_pow_mul_integral Set.preimage_image_preimage h₂,
    normAtAllPlaces_image_preimage_expMapBasis, setLIntegral_congr (compactSet_ae K),
    setLIntegral_expMapBasis_image measurableSet_closure (by fun_prop),
    setLIntegral_expMapBasis_image measurableSet_interior (by fun_prop),
    setLIntegral_congr (closure_paramSet_ae_interior K)]

Premises used: measurableSet_closure, measurableSet_interior, NumberField.mixedEmbedding.volume_eq_two_pow_mul_two_pi_pow_mul_integral, NumberField.mixedEmbedding.normAtAllPlaces_image_preimage_of_nonneg, NumberField.mixedEmbedding.fundamentalCone.setLIntegral_expMapBasis_image, NumberField.mixedEmbedding.fundamentalCone.nonneg_of_mem_compactSet, NumberField.mixedEmbedding.fundamentalCone.compactSet_ae, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_image_preimage_expMapBasis, Set.preimage_image_preimage, MeasureTheory.setLIntegral_congr, NumberField.mixedEmbedding.fundamentalCone.closure_paramSet_ae_interior