-- Mathlib/MeasureTheory/Function/Jacobian.lean
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)

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t

Tactic:
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

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
this : ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t
⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t

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

Tactic:
have hf' : ApproximatesLinearOn f (B : E →L[ℝ] E) s δ := by convert hf

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

Proof state:
case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
this : ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t
δ₀ : ℝ≥0
h : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
h' : 0 < δ₀
⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t

Tactic:
exact ⟨δ₀, h', h⟩

Result:
no goals

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

Tactic:
refine Tendsto.mul (tendsto_const_nhds.mul ?_) tendsto_id

Result:
case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
H : ‖↑B.symm‖₊ ≠ 0
⊢ Tendsto (fun δ => (‖↑B.symm‖₊⁻¹ - δ)⁻¹) (𝓝 0) (𝓝 (‖↑B.symm‖₊⁻¹ - 0)⁻¹)

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
⊢ Tendsto (fun δ => ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0) (𝓝 (‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - 0)⁻¹ * 0))

Tactic:
rcases eq_or_ne ‖(B.symm : E →L[ℝ] E)‖₊ 0 with (H | H)

Result:
case inl
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
H : ‖↑B.symm‖₊ = 0
⊢ Tendsto (fun δ => ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0) (𝓝 (‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - 0)⁻¹ * 0))

case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
H : ‖↑B.symm‖₊ ≠ 0
⊢ Tendsto (fun δ => ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0) (𝓝 (‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - 0)⁻¹ * 0))