/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  have hA : A.det ≠ 0 := by
    intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
  -- let `B` be the continuous linear equiv version of `A`.
  let B := A.toContinuousLinearEquivOfDetNeZero hA
  -- the determinant of `B.symm` is bounded by `m⁻¹`
  have I : ENNReal.ofReal |(B.symm : E →L[ℝ] E).det| < (m⁻¹ : ℝ≥0) := by
    simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,
      ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm ⊢
    exact NNReal.inv_lt_inv mpos.ne' hm
  -- therefore, we may apply `addHaar_image_le_mul_of_det_lt` to `B.symm` and `m⁻¹`.
  obtain ⟨δ₀, δ₀pos, hδ₀⟩ :
    ∃ δ : ℝ≥0,
      0 < δ ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g (B.symm : E →L[ℝ] E) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t := by
    have :
      ∀ᶠ δ : ℝ≥0 in 𝓝[>] 0,
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g (B.symm : E →L[ℝ] E) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t :=
      addHaar_image_le_mul_of_det_lt μ B.symm I
    rcases (this.and self_mem_nhdsWithin).exists with ⟨δ₀, h, h'⟩
    exact ⟨δ₀, h', h⟩
  -- record smallness conditions for `δ` that will be needed to apply `hδ₀` below.
  have L1 : ∀ᶠ δ in 𝓝 (0 : ℝ≥0), Subsingleton E ∨ δ < ‖(B.symm : E →L[ℝ] E)‖₊⁻¹ := by
    by_cases h : Subsingleton E
    · simp only [h, true_or_iff, eventually_const]
    simp only [h, false_or_iff]
    apply Iio_mem_nhds
    simpa only [h, false_or_iff, inv_pos] using B.subsingleton_or_nnnorm_symm_pos
  have L2 :
    ∀ᶠ δ in 𝓝 (0 : ℝ≥0), ‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - δ)⁻¹ * δ < δ₀ := by
    have :
      Tendsto (fun δ => ‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0)
        (𝓝 (‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - 0)⁻¹ * 0)) := by
      rcases eq_or_ne ‖(B.symm : E →L[ℝ] E)‖₊ 0 with (H | H)
      · simpa only [H, zero_mul] using tendsto_const_nhds
      refine' Tendsto.mul (tendsto_const_nhds.mul _) tendsto_id
      refine' (Tendsto.sub tendsto_const_nhds tendsto_id).inv₀ _
      simpa only [tsub_zero, inv_eq_zero, Ne.def] using H
    simp only [mul_zero] at this
    exact (tendsto_order.1 this).2 δ₀ δ₀pos
  -- let `δ` be small enough, and `f` approximated by `B` up to `δ`.
  filter_upwards [L1, L2]
  intro δ h1δ h2δ s f hf
  have hf' : ApproximatesLinearOn f (B : E →L[ℝ] E) s δ := by convert hf
  let F := hf'.toPartialEquiv h1δ
  -- the condition to be checked can be reformulated in terms of the inverse maps
  suffices H : μ (F.symm '' F.target) ≤ (m⁻¹ : ℝ≥0) * μ F.target
  · change (m : ℝ≥0∞) * μ F.source ≤ μ F.target
    rwa [← F.symm_image_target_eq_source, mul_comm, ← ENNReal.le_div_iff_mul_le, div_eq_mul_inv,
      mul_comm, ← ENNReal.coe_inv mpos.ne']
    · apply Or.inl
      simpa only [ENNReal.coe_eq_zero, Ne.def] using mpos.ne'
    · simp only [ENNReal.coe_ne_top, true_or_iff, Ne.def, not_false_iff]
  -- as `f⁻¹` is well approximated by `B⁻¹`, the conclusion follows from `hδ₀`
  -- and our choice of `δ`.
  

Proof state:
case H
E : Type u_1
F✝ : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F✝
inst✝³ : NormedSpace ℝ F✝
s✝ : Set E
f✝ : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
hA : ContinuousLinearMap.det A ≠ 0
B : E ≃L[ℝ] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA
I : ENNReal.ofReal |ContinuousLinearMap.det ↑(ContinuousLinearEquiv.symm B)| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ :
  ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑(ContinuousLinearEquiv.symm B)) t δ₀ → ↑↑μ (g '' t) ≤ ↑m⁻¹ * ↑↑μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑(ContinuousLinearEquiv.symm B)‖₊⁻¹
L2 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, ‖↑(ContinuousLinearEquiv.symm B)‖₊ * (‖↑(ContinuousLinearEquiv.symm B)‖₊⁻¹ - δ)⁻¹ * δ < δ₀
δ : ℝ≥0
h1δ : Subsingleton E ∨ δ < ‖↑(ContinuousLinearEquiv.symm B)‖₊⁻¹
h2δ : ‖↑(ContinuousLinearEquiv.symm B)‖₊ * (‖↑(ContinuousLinearEquiv.symm B)‖₊⁻¹ - δ)⁻¹ * δ < δ₀
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hf' : ApproximatesLinearOn f (↑B) s δ
F : PartialEquiv E E := ApproximatesLinearOn.toPartialEquiv hf' h1δ
⊢ ↑↑μ (↑(PartialEquiv.symm F) '' F.target) ≤ ↑m⁻¹ * ↑↑μ F.target

Next tactic:
exact hδ₀ _ _ ((hf'.to_inv h1δ).mono_num h2δ.le)