Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑isoG.symm) μ
eg : eP → eG → eE' := fun ep ex ↦ isoE'.symm (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF := ↑(isoE'.arrowCongr isoF).symm ∘SL L
R : eP × eG → eF := fun q ↦ (ef ⋆[eL, eμ] eg q.1) q.2
⊢ ContDiffOn 𝕜 (↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
have R_contdiff : ContDiffOn 𝕜 n R ((isoP ⁻¹' s) ×ˢ univ) := by
    have hek : IsCompact (isoG ⁻¹' k) := isoG.toHomeomorph.isClosedEmbedding.isCompact_preimage hk
    have hes : IsOpen (isoP ⁻¹' s) := isoP.continuous.isOpen_preimage _ hs
    refine contDiffOn_convolution_right_with_param_aux eL hes hek ?_ ?_ ?_
    · intro p x hp hx
      simp only [eg,
        ContinuousLinearEquiv.map_eq_zero_iff]
      exact hgs _ _ hp hx
    · exact (locallyIntegrable_map_homeomorph isoG.symm.toHomeomorph).2 hf
    · apply isoE'.symm.contDiff.comp_contDiffOn
      apply hg.comp (isoP.prodCongr isoG).contDiff.contDiffOn
      rintro ⟨p, x⟩ ⟨hp, -⟩
      simpa only [mem_preimage, ContinuousLinearEquiv.prodCongr_apply, prodMk_mem_set_prod_eq,
        mem_univ, and_true] using hp

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑isoG.symm) μ
eg : eP → eG → eE' := fun ep ex ↦ isoE'.symm (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF := ↑(isoE'.arrowCongr isoF).symm ∘SL L
R : eP × eG → eF := fun q ↦ (ef ⋆[eL, eμ] eg q.1) q.2
R_contdiff : ContDiffOn 𝕜 (↑n) R ((⇑isoP ⁻¹' s) ×ˢ univ)
⊢ ContDiffOn 𝕜 (↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)