Theorem: sard_boundaryless
Proof state before:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst✝¹² : MeasurableSpace E
inst✝¹¹ : BorelSpace E
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
G : Type u_5
inst✝⁸ : TopologicalSpace G
J : ModelWithCorners ℝ F G
N : Type u_6
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ChartedSpace G N
inst✝⁵ : SmoothManifoldWithCorners J N
inst✝⁴ : FiniteDimensional ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
m n r✝ : ℕ
hm : finrank ℝ E = m
hn : finrank ℝ F = n
hr✝ : r✝ > m - n
f : M → N
hf : ContMDiff I J (↑r✝) f
inst✝¹ : I.Boundaryless
inst✝ : J.Boundaryless
f' : (x : M) → TangentSpace I x →L[ℝ] TangentSpace J (f x)
s : Set M
hf' : ∀ x ∈ s, HasMFDerivAt I J f x (f' x)
h'f' : ∀ x ∈ s, ¬Surjective ⇑(f' x)
x✝ : M
e : PartialHomeomorph M H := chartAt H x✝
μ : Measure F
hμ : μ.IsAddHaarMeasure
y : N
e' : PartialHomeomorph N G := chartAt G y
w : Set E := ↑(e.extend I) '' (e.source ∩ f ⁻¹' e'.source)
s_better : Set E := ↑(e.extend I) '' (s ∩ e.source ∩ f ⁻¹' e'.source)
f_local : E → F := (↑J ∘ ↑e') ∘ f ∘ ↑(e.extend I).symm
hwopen : IsOpen w
hsw : s_better ⊆ w
hsbetter₀ : s_better ⊆ ↑(e.extend I) '' e.source
cor : ↑(e.extend I).symm ∘ ↑(e.extend I) '' (s ∩ e.source ∩ f ⁻¹' e'.source) = s ∩ e.source ∩ f ⁻¹' e'.source
hsbetter : ↑(e.extend I).symm '' s_better = s ∩ e.source ∩ f ⁻¹' e'.source
hsbetter₁ : ↑(e.extend I).symm '' s_better ⊆ s
hsbetter₂ : ↑(e.extend I).symm '' s_better ⊆ e.source
hw : f ∘ ↑(e.extend I).symm '' w ⊆ e'.source
hsbetter₃ : f ∘ ↑(e.extend I).symm '' s_better ⊆ e'.source
this✝¹ : ↑J ∘ ↑e' '' (e'.source ∩ f '' (e.source ∩ s)) = f_local '' s_better
x : E
hx : x ∈ s_better
D : TangentSpace 𝓘(ℝ, E) x →L[ℝ] TangentSpace 𝓘(ℝ, F) (f_local x) := mfderiv 𝓘(ℝ, E) 𝓘(ℝ, F) f_local x
A : TangentSpace 𝓘(ℝ, E) x →L[ℝ] TangentSpace I (↑(e.extend I).symm x) := mfderiv 𝓘(ℝ, E) I (↑(e.extend I).symm) x
B : TangentSpace I (↑(e.extend I).symm x) →L[ℝ] TangentSpace J (f (↑(e.extend I).symm x)) :=
  mfderiv I J f (↑(e.extend I).symm x)
C : TangentSpace J ((f ∘ ↑(e.extend I).symm) x) →L[ℝ]
  TangentSpace 𝓘(ℝ, F) (↑(e'.extend J) ((f ∘ ↑(e.extend I).symm) x)) :=
  mfderiv J 𝓘(ℝ, F) (↑(e'.extend J)) ((f ∘ ↑(e.extend I).symm) x)
hnA : ↑(e.extend I) '' e.source ∈ 𝓝 x
hr : 1 ≤ ↑r✝
he : e ∈ maximalAtlas I M
this✝ : ↑I '' e.target ∈ 𝓝 x
hA✝ : MDifferentiableAt 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x
hB : MDifferentiableAt I J f ((e.invFun ∘ I.invFun) x)
hBA : MDifferentiableAt 𝓘(ℝ, E) J (f ∘ e.invFun ∘ I.invFun) x
hnC : f ∘ e.invFun ∘ I.invFun '' w ∈ 𝓝 ((f ∘ e.invFun ∘ I.invFun) x)
hC : MDifferentiableAt J 𝓘(ℝ, F) (↑J ∘ ↑e') ((f ∘ e.invFun ∘ I.invFun) x)
comp : TangentSpace 𝓘(ℝ, E) x →L[ℝ] TangentSpace 𝓘(ℝ, F) (↑(e'.extend J) ((f ∘ ↑(e.extend I).symm) x)) :=
  C.comp (B.comp A)
r : comp = D :=
  Trans.trans
    (Trans.trans
      (Trans.trans
        (Trans.trans
          (Trans.trans (Trans.trans (Trans.trans rfl rfl) rfl)
            (Eq.mpr
              (id
                (congrArg
                  (fun _a =>
                    C.comp
                        ((mfderiv I J f ((e.invFun ∘ I.invFun) x)).comp (mfderiv 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x)) =
                      C.comp _a)
                  (mfderiv_comp x hB hA✝)))
              (Eq.refl
                (C.comp ((mfderiv I J f ((e.invFun ∘ I.invFun) x)).comp (mfderiv 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x))))))
          rfl)
        (Eq.mpr
          (id
            (congrArg
              (fun _a =>
                (mfderiv J 𝓘(ℝ, F) (↑J ∘ ↑e') ((f ∘ e.invFun ∘ I.invFun) x)).comp
                    (mfderiv 𝓘(ℝ, E) J (f ∘ e.invFun ∘ I.invFun) x) =
                  _a)
              (mfderiv_comp x hC hBA)))
          (Eq.refl
            ((mfderiv J 𝓘(ℝ, F) (↑J ∘ ↑e') ((f ∘ e.invFun ∘ I.invFun) x)).comp
              (mfderiv 𝓘(ℝ, E) J (f ∘ e.invFun ∘ I.invFun) x)))))
      rfl)
    rfl
x' : M := ↑(e.extend I).symm x
aux : x' ∈ s
this : f' x' = B
hBsurj : ¬Surjective ⇑B
hA : Bijective ⇑A
⊢ ¬Surjective ⇑D
Tactic:
have hC : Bijective C :=
  extendedChart_differential_bijective J (chart_mem_atlas G _) (hsbetter₃ (mem_image_of_mem _ hx))
Proof state after:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst✝¹² : MeasurableSpace E
inst✝¹¹ : BorelSpace E
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
G : Type u_5
inst✝⁸ : TopologicalSpace G
J : ModelWithCorners ℝ F G
N : Type u_6
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ChartedSpace G N
inst✝⁵ : SmoothManifoldWithCorners J N
inst✝⁴ : FiniteDimensional ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
m n r✝ : ℕ
hm : finrank ℝ E = m
hn : finrank ℝ F = n
hr✝ : r✝ > m - n
f : M → N
hf : ContMDiff I J (↑r✝) f
inst✝¹ : I.Boundaryless
inst✝ : J.Boundaryless
f' : (x : M) → TangentSpace I x →L[ℝ] TangentSpace J (f x)
s : Set M
hf' : ∀ x ∈ s, HasMFDerivAt I J f x (f' x)
h'f' : ∀ x ∈ s, ¬Surjective ⇑(f' x)
x✝ : M
e : PartialHomeomorph M H := chartAt H x✝
μ : Measure F
hμ : μ.IsAddHaarMeasure
y : N
e' : PartialHomeomorph N G := chartAt G y
w : Set E := ↑(e.extend I) '' (e.source ∩ f ⁻¹' e'.source)
s_better : Set E := ↑(e.extend I) '' (s ∩ e.source ∩ f ⁻¹' e'.source)
f_local : E → F := (↑J ∘ ↑e') ∘ f ∘ ↑(e.extend I).symm
hwopen : IsOpen w
hsw : s_better ⊆ w
hsbetter₀ : s_better ⊆ ↑(e.extend I) '' e.source
cor : ↑(e.extend I).symm ∘ ↑(e.extend I) '' (s ∩ e.source ∩ f ⁻¹' e'.source) = s ∩ e.source ∩ f ⁻¹' e'.source
hsbetter : ↑(e.extend I).symm '' s_better = s ∩ e.source ∩ f ⁻¹' e'.source
hsbetter₁ : ↑(e.extend I).symm '' s_better ⊆ s
hsbetter₂ : ↑(e.extend I).symm '' s_better ⊆ e.source
hw : f ∘ ↑(e.extend I).symm '' w ⊆ e'.source
hsbetter₃ : f ∘ ↑(e.extend I).symm '' s_better ⊆ e'.source
this✝¹ : ↑J ∘ ↑e' '' (e'.source ∩ f '' (e.source ∩ s)) = f_local '' s_better
x : E
hx : x ∈ s_better
D : TangentSpace 𝓘(ℝ, E) x →L[ℝ] TangentSpace 𝓘(ℝ, F) (f_local x) := mfderiv 𝓘(ℝ, E) 𝓘(ℝ, F) f_local x
A : TangentSpace 𝓘(ℝ, E) x →L[ℝ] TangentSpace I (↑(e.extend I).symm x) := mfderiv 𝓘(ℝ, E) I (↑(e.extend I).symm) x
B : TangentSpace I (↑(e.extend I).symm x) →L[ℝ] TangentSpace J (f (↑(e.extend I).symm x)) :=
  mfderiv I J f (↑(e.extend I).symm x)
C : TangentSpace J ((f ∘ ↑(e.extend I).symm) x) →L[ℝ]
  TangentSpace 𝓘(ℝ, F) (↑(e'.extend J) ((f ∘ ↑(e.extend I).symm) x)) :=
  mfderiv J 𝓘(ℝ, F) (↑(e'.extend J)) ((f ∘ ↑(e.extend I).symm) x)
hnA : ↑(e.extend I) '' e.source ∈ 𝓝 x
hr : 1 ≤ ↑r✝
he : e ∈ maximalAtlas I M
this✝ : ↑I '' e.target ∈ 𝓝 x
hA✝ : MDifferentiableAt 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x
hB : MDifferentiableAt I J f ((e.invFun ∘ I.invFun) x)
hBA : MDifferentiableAt 𝓘(ℝ, E) J (f ∘ e.invFun ∘ I.invFun) x
hnC : f ∘ e.invFun ∘ I.invFun '' w ∈ 𝓝 ((f ∘ e.invFun ∘ I.invFun) x)
hC✝ : MDifferentiableAt J 𝓘(ℝ, F) (↑J ∘ ↑e') ((f ∘ e.invFun ∘ I.invFun) x)
comp : TangentSpace 𝓘(ℝ, E) x →L[ℝ] TangentSpace 𝓘(ℝ, F) (↑(e'.extend J) ((f ∘ ↑(e.extend I).symm) x)) :=
  C.comp (B.comp A)
r : comp = D :=
  Trans.trans
    (Trans.trans
      (Trans.trans
        (Trans.trans
          (Trans.trans (Trans.trans (Trans.trans rfl rfl) rfl)
            (Eq.mpr
              (id
                (congrArg
                  (fun _a =>
                    C.comp
                        ((mfderiv I J f ((e.invFun ∘ I.invFun) x)).comp (mfderiv 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x)) =
                      C.comp _a)
                  (mfderiv_comp x hB hA✝)))
              (Eq.refl
                (C.comp ((mfderiv I J f ((e.invFun ∘ I.invFun) x)).comp (mfderiv 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x))))))
          rfl)
        (Eq.mpr
          (id
            (congrArg
              (fun _a =>
                (mfderiv J 𝓘(ℝ, F) (↑J ∘ ↑e') ((f ∘ e.invFun ∘ I.invFun) x)).comp
                    (mfderiv 𝓘(ℝ, E) J (f ∘ e.invFun ∘ I.invFun) x) =
                  _a)
              (mfderiv_comp x hC✝ hBA)))
          (Eq.refl
            ((mfderiv J 𝓘(ℝ, F) (↑J ∘ ↑e') ((f ∘ e.invFun ∘ I.invFun) x)).comp
              (mfderiv 𝓘(ℝ, E) J (f ∘ e.invFun ∘ I.invFun) x)))))
      rfl)
    rfl
x' : M := ↑(e.extend I).symm x
aux : x' ∈ s
this : f' x' = B
hBsurj : ¬Surjective ⇑B
hA : Bijective ⇑A
hC : Bijective ⇑C
⊢ ¬Surjective ⇑D