/-- The image of an additive Haar measure under a surjective linear map is proportional to a given
additive Haar measure. The proportionality factor will be infinite if the linear map has a
nontrivial kernel. -/
theorem LinearMap.exists_map_addHaar_eq_smul_addHaar' (h : Function.Surjective L) :
    ∃ (c : ℝ≥0∞), 0 < c ∧ c < ∞ ∧ μ.map L = (c * addHaar (univ : Set (LinearMap.ker L))) • ν := by
  /- This is true for the second projection in product spaces, as the projection of the Haar
  measure `μS.prod μT` is equal to the Haar measure `μT` multiplied by the total mass of `μS`. This
  is also true for linear equivalences, as they map Haar measure to Haar measure. The general case
  follows from these two and linear algebra, as `L` can be interpreted as the composition of the
  projection `P` on a complement `T` to its kernel `S`, together with a linear equivalence. -/
  have : ProperSpace E := properSpace_of_locallyCompactSpace 𝕜
  have : FiniteDimensional 𝕜 E := finiteDimensional_of_locallyCompactSpace 𝕜
  have : ProperSpace F := by
    rcases subsingleton_or_nontrivial E with hE|hE
    · have : Subsingleton F := Function.Surjective.subsingleton h
      infer_instance
    · have : ProperSpace 𝕜 := properSpace_of_locallyCompact_module 𝕜 E
      have : FiniteDimensional 𝕜 F := Module.Finite.of_surjective L h
      exact FiniteDimensional.proper 𝕜 F
  let S : Submodule 𝕜 E := LinearMap.ker L
  obtain ⟨T, hT⟩ : ∃ T : Submodule 𝕜 E, IsCompl S T := Submodule.exists_isCompl S
  let M : (S × T) ≃ₗ[𝕜] E := Submodule.prodEquivOfIsCompl S T hT
  have M_cont : Continuous M.symm := LinearMap.continuous_of_finiteDimensional _
  let P : S × T →ₗ[𝕜] T := LinearMap.snd 𝕜 S T
  have P_cont : Continuous P := LinearMap.continuous_of_finiteDimensional _
  have I : Function.Bijective (LinearMap.domRestrict L T) :=
    ⟨LinearMap.injective_domRestrict_iff.2 (IsCompl.inf_eq_bot hT.symm),
    (LinearMap.surjective_domRestrict_iff h).2 hT.symm.sup_eq_top⟩
  let L' : T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (LinearMap.domRestrict L T) I
  have L'_cont : Continuous L' := LinearMap.continuous_of_finiteDimensional _
  have A : L = (L' : T →ₗ[𝕜] F).comp (P.comp (M.symm : E →ₗ[𝕜] (S × T))) := by
    ext x
    obtain ⟨y, z, hyz⟩ : ∃ (y : S) (z : T), M.symm x = (y, z) := ⟨_, _, rfl⟩
    have : x = M (y, z) := by
      rw [← hyz]; simp only [LinearEquiv.apply_symm_apply]
    simp [this]
  have I : μ.map L = ((μ.map M.symm).map P).map L' := by
    rw [Measure.map_map, Measure.map_map, A]
    · rfl
    · exact L'_cont.measurable.comp P_cont.measurable
    · exact M_cont.measurable
    · exact L'_cont.measurable
    · exact P_cont.measurable
  let μS : Measure S := addHaar
  let μT : Measure T := addHaar
  obtain ⟨c₀, c₀_pos, c₀_fin, h₀⟩ :
      ∃ c₀ : ℝ≥0∞, c₀ ≠ 0 ∧ c₀ ≠ ∞ ∧ μ.map M.symm = c₀ • μS.prod μT := by
    have : IsAddHaarMeasure (μ.map M.symm) :=
      M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map μ
    exact isAddHaarMeasure_eq_smul _ _
  

Proof state:
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : IsAddHaarMeasure μ
inst✝¹ : IsAddHaarMeasure ν
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := Submodule.prodEquivOfIsCompl S T hT
M_cont : Continuous ⇑(LinearEquiv.symm M)
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(domRestrict L T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (domRestrict L T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑(LinearEquiv.symm M)
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑(LinearEquiv.symm M)) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑(LinearEquiv.symm M)) μ = c₀ • Measure.prod μS μT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * ↑↑addHaar univ) • ν

Next tactic:
have J : (μS.prod μT).map P = (μS univ) • μT := map_snd_prod