-- Module: Mathlib.NumberTheory.LSeries.AbstractFuncEq
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℂ E
P : WeakFEPair E
inst_2 : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - (↑P.k : ℂ) - 1).re
_ :
  Eq.{u_1 + 1} (α := E)
    (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioi 0)) fun (x : ℝ) ↦
      (↑x : ℂ) ^ (s - 1) •
        HAdd.hAdd (α := E) (indicator (Ioo 0 1) (fun (t : ℝ) ↦ P.f₀ - (P.ε * (↑(t ^ (-P.k)) : ℂ)) • P.g₀) x)
          (indicator {1} (fun (x : ℝ) ↦ P.f₀ - P.f 1) x))
    (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioi 0)) fun (x : ℝ) ↦
      HSMul.hSMul (β := E) ((↑x : ℂ) ^ (s - 1))
        (indicator (Ioo 0 1) (fun (t : ℝ) ↦ P.f₀ - (P.ε * (↑(t ^ (-P.k)) : ℂ)) • P.g₀) x))
_₁ :
  Eq.{u_1 + 1} (α := E)
    (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioi 0)) fun (x : ℝ) ↦
      HSMul.hSMul (β := E) ((↑x : ℂ) ^ (s - 1))
        (indicator (Ioo 0 1) (fun (t : ℝ) ↦ P.f₀ - (P.ε * (↑(t ^ (-P.k)) : ℂ)) • P.g₀) x))
    (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
      (↑x : ℂ) ^ (s - 1) • (P.f₀ - (P.ε * (↑(x ^ (-P.k)) : ℂ)) • P.g₀))
_₂ :
  Eq.{u_1 + 1} (α := E)
    (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
      (↑x : ℂ) ^ (s - 1) • (P.f₀ - (P.ε * (↑(x ^ (-P.k)) : ℂ)) • P.g₀))
    (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
      (↑x : ℂ) ^ (s - 1) • P.f₀ - P.ε • (↑x : ℂ) ^ (s - (↑P.k : ℂ) - 1) • P.g₀)
_₃ :
  (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
      (↑x : ℂ) ^ (s - 1) • P.f₀ - P.ε • (↑x : ℂ) ^ (s - (↑P.k : ℂ) - 1) • P.g₀) =
    HSMul.hSMul (α := ℂ)
        (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
          (↑x : ℂ) ^ (s - 1))
        P.f₀ -
      P.ε •
        HSMul.hSMul (α := ℂ)
          (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
            (↑x : ℂ) ^ (s - (↑P.k : ℂ) - 1))
          P.g₀
_₄ :
  HSMul.hSMul (α := ℂ)
        (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
          (↑x : ℂ) ^ (s - 1))
        P.f₀ -
      P.ε •
        HSMul.hSMul (α := ℂ)
          (@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Ioc 0 1)) fun (x : ℝ) ↦
            (↑x : ℂ) ^ (s - (↑P.k : ℂ) - 1))
          P.g₀ =
    (1 / s) • P.f₀ + (P.ε / ((↑P.k : ℂ) - s)) • P.g₀
⊢ mellin (fun (x : ℝ) ↦ P.f_modif x - P.f x + P.f₀) s = (1 / s) • P.f₀ + (P.ε / ((↑P.k : ℂ) - s)) • P.g₀

Tactic:
calc
  _ = ∫ (x : ℝ) in Ioi 0, (x : ℂ) ^ (s - 1) •
      ((Ioo 0 1).indicator (fun t : ℝ ↦ P.f₀ - (P.ε * ↑(t ^ (-P.k))) • P.g₀) x
      + ({1} : Set ℝ).indicator (fun _ ↦ P.f₀ - P.f 1) x) :=
    setIntegral_congr_fun measurableSet_Ioi (fun x hx ↦ by simp [f_modif_aux1 P hx])
  _ = ∫ (x : ℝ) in Ioi 0, (x : ℂ) ^ (s - 1) • ((Ioo 0 1).indicator
      (fun t : ℝ ↦ P.f₀ - (P.ε * ↑(t ^ (-P.k))) • P.g₀) x) := by
    refine setIntegral_congr_ae measurableSet_Ioi (eventually_of_mem (U := {1}ᶜ)
        (compl_mem_ae_iff.mpr (subsingleton_singleton.measure_zero _)) (fun x hx _ ↦ ?_))
    rw [indicator_of_notMem hx, add_zero]
  _ = ∫ (x : ℝ) in Ioc 0 1, (x : ℂ) ^ (s - 1) • (P.f₀ - (P.ε * ↑(x ^ (-P.k))) • P.g₀) := by
    simp_rw [← indicator_smul, setIntegral_indicator measurableSet_Ioo,
      inter_eq_right.mpr Ioo_subset_Ioi_self, integral_Ioc_eq_integral_Ioo]
  _ = ∫ x : ℝ in Ioc 0 1, ((x : ℂ) ^ (s - 1) • P.f₀ - P.ε • (x : ℂ) ^ (s - P.k - 1) • P.g₀) := by
    refine setIntegral_congr_fun measurableSet_Ioc (fun x ⟨hx, _⟩ ↦ ?_)
    rw [ofReal_cpow hx.le, ofReal_neg, smul_sub, ← mul_smul, mul_comm, mul_assoc, mul_smul,
      mul_comm, ← cpow_add _ _ (ofReal_ne_zero.mpr hx.ne'), ← sub_eq_add_neg, sub_right_comm]
  _ = (∫ (x : ℝ) in Ioc 0 1, (x : ℂ) ^ (s - 1)) • P.f₀
        - P.ε • (∫ (x : ℝ) in Ioc 0 1, (x : ℂ) ^ (s - P.k - 1)) • P.g₀ := by
    rw [integral_sub, integral_smul, integral_smul_const, integral_smul_const]
    · apply Integrable.smul_const
      rw [← IntegrableOn, ← intervalIntegrable_iff_integrableOn_Ioc_of_le zero_le_one]
      exact intervalIntegral.intervalIntegrable_cpow' h_re1
    · refine (Integrable.smul_const ?_ _).smul _
      rw [← IntegrableOn, ← intervalIntegrable_iff_integrableOn_Ioc_of_le zero_le_one]
      exact intervalIntegral.intervalIntegrable_cpow' h_re2
  _ = _ := by simp_rw [← intervalIntegral.integral_of_le zero_le_one,
      integral_cpow (Or.inl h_re1), integral_cpow (Or.inl h_re2), ofReal_zero, ofReal_one,
      one_cpow, sub_add_cancel, zero_cpow fun h ↦ lt_irrefl _ (P.hk.le.trans_lt (zero_re ▸ h ▸ hs)),
      zero_cpow (sub_ne_zero.mpr (fun h ↦ lt_irrefl _ ((ofReal_re _) ▸ h ▸ hs)) : s - P.k ≠ 0),
      sub_zero, sub_eq_add_neg (_ • _), ← mul_smul, ← neg_smul, mul_one_div, ← div_neg, neg_sub]

Premises used: Set.Ioc, Set.Ioi, HMul.hMul, HSub.hSub, WeakFEPair.ε, MeasureTheory.Measure.restrict, WeakFEPair.g₀, measurableSet_Ioi, WeakFEPair.f₀, Complex.ofReal, MeasureTheory.setIntegral_congr_fun, HPow.hPow, MeasureTheory.MeasureSpace.volume, MeasureTheory.integral, WeakFEPair.k, HSMul.hSMul, OfNat.ofNat