l3lab/ntp-mathlib huggingface dataset:
theorem hasMFDerivAt_fst (x : M × M') :
    HasMFDerivAt (I.prod I') I Prod.fst x
      (ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) := by
  refine' ⟨continuous_fst.continuousAt, _⟩
  have :
    ∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
      (extChartAt I x.1 ∘ Prod.fst ∘ (extChartAt (I.prod I') x).symm) y = y.1 := by
    /- porting note: was
    apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
    mfld_set_tac
    -/
    filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy
    rw [extChartAt_prod] at hy
    exact (extChartAt I x.1).right_inv hy.1
  apply HasFDerivWithinAt.congr_of_eventuallyEq hasFDerivWithinAt_fst this
  -- porting note: next line was `simp only [mfld_simps]`
  

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x✝ : M
x : M × M'
this :
  ∀ᶠ (y : E × E') in 𝓝[range ↑(ModelWithCorners.prod I I')] ↑(extChartAt (ModelWithCorners.prod I I') x) x,
    (↑(extChartAt I x.1) ∘ Prod.fst ∘ ↑(PartialEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.1
⊢ (↑(extChartAt I x.1) ∘ Prod.fst ∘ ↑(PartialEquiv.symm (extChartAt (ModelWithCorners.prod I I') x)))
      (↑(extChartAt (ModelWithCorners.prod I I') x) x) =
    (↑(extChartAt (ModelWithCorners.prod I I') x) x).1

Next tactic:
exact (extChartAt I x.1).right_inv <| (extChartAt I x.1).map_source (mem_extChartAt_source _ _)
theorem mfderivWithin_fst {s : Set (M × M')} {x : M × M'}
    (hxs : UniqueMDiffWithinAt (I.prod I') s x) :
    mfderivWithin (I.prod I') I Prod.fst s x =
      ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) :=
  by 

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x✝ : M
s : Set (M × M')
x : M × M'
hxs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s x
⊢ mfderivWithin (ModelWithCorners.prod I I') I Prod.fst s x =
    ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)

Next tactic:
rw [MDifferentiable.mfderivWithin (mdifferentiableAt_fst I I') hxs]
theorem mfderivWithin_fst {s : Set (M × M')} {x : M × M'}
    (hxs : UniqueMDiffWithinAt (I.prod I') s x) :
    mfderivWithin (I.prod I') I Prod.fst s x =
      ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) :=
  by rw [MDifferentiable.mfderivWithin (mdifferentiableAt_fst I I') hxs]; 

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x✝ : M
s : Set (M × M')
x : M × M'
hxs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s x
⊢ mfderiv (ModelWithCorners.prod I I') I Prod.fst x =
    ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)

Next tactic:
exact mfderiv_fst I I'
@[simp, mfld_simps]
theorem tangentMap_prod_fst {p : TangentBundle (I.prod I') (M × M')} :
    tangentMap (I.prod I') I Prod.fst p = ⟨p.proj.1, p.2.1⟩ := by
  -- porting note: `rfl` wasn't needed
  

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x : M
p : TangentBundle (ModelWithCorners.prod I I') (M × M')
⊢ tangentMap (ModelWithCorners.prod I I') I Prod.fst p = { proj := p.proj.1, snd := p.snd.1 }

Next tactic:
simp [tangentMap]
@[simp, mfld_simps]
theorem tangentMap_prod_fst {p : TangentBundle (I.prod I') (M × M')} :
    tangentMap (I.prod I') I Prod.fst p = ⟨p.proj.1, p.2.1⟩ := by
  -- porting note: `rfl` wasn't needed
  simp [tangentMap]; 

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x : M
p : TangentBundle (ModelWithCorners.prod I I') (M × M')
⊢ (ContinuousLinearMap.fst 𝕜 (TangentSpace I p.proj.1) (TangentSpace I' p.proj.2)) p.snd = p.snd.1

Next tactic:
rfl
theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by
  

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x : M
s : Set (M × M')
p : TangentBundle (ModelWithCorners.prod I I') (M × M')
hs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj
⊢ tangentMapWithin (ModelWithCorners.prod I I') I Prod.fst s p = { proj := p.proj.1, snd := p.snd.1 }

Next tactic:
simp only [tangentMapWithin]
theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by
  simp only [tangentMapWithin]
  

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x : M
s : Set (M × M')
p : TangentBundle (ModelWithCorners.prod I I') (M × M')
hs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj
⊢ { proj := p.proj.1, snd := (mfderivWithin (ModelWithCorners.prod I I') I Prod.fst s p.proj) p.snd } =
    { proj := p.proj.1, snd := p.snd.1 }

Next tactic:
rw [mfderivWithin_fst]
theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by
  simp only [tangentMapWithin]
  rw [mfderivWithin_fst]
  · 

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x : M
s : Set (M × M')
p : TangentBundle (ModelWithCorners.prod I I') (M × M')
hs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj
⊢ { proj := p.proj.1, snd := (ContinuousLinearMap.fst 𝕜 (TangentSpace I p.proj.1) (TangentSpace I' p.proj.2)) p.snd } =
    { proj := p.proj.1, snd := p.snd.1 }

Next tactic:
rcases p with ⟨⟩
theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by
  simp only [tangentMapWithin]
  rw [mfderivWithin_fst]
  · rcases p with ⟨⟩; 

Proof state:
case mk
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x : M
s : Set (M × M')
proj✝ : M × M'
snd✝ : TangentSpace (ModelWithCorners.prod I I') proj✝
hs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s { proj := proj✝, snd := snd✝ }.proj
⊢ { proj := { proj := proj✝, snd := snd✝ }.proj.1,
      snd :=
        (ContinuousLinearMap.fst 𝕜 (TangentSpace I { proj := proj✝, snd := snd✝ }.proj.1)
            (TangentSpace I' { proj := proj✝, snd := snd✝ }.proj.2))
          { proj := proj✝, snd := snd✝ }.snd } =
    { proj := { proj := proj✝, snd := snd✝ }.proj.1, snd := { proj := proj✝, snd := snd✝ }.snd.1 }

Next tactic:
rfl
theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by
  simp only [tangentMapWithin]
  rw [mfderivWithin_fst]
  · rcases p with ⟨⟩; rfl
  · 

Proof state:
case hxs
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s✝ : Set M
x : M
s : Set (M × M')
p : TangentBundle (ModelWithCorners.prod I I') (M × M')
hs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj
⊢ UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj

Next tactic:
exact hs
theorem hasMFDerivAt_snd (x : M × M') :
    HasMFDerivAt (I.prod I') I' Prod.snd x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) := by
  

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x✝ : M
x : M × M'
⊢ HasMFDerivAt (ModelWithCorners.prod I I') I' Prod.snd x
    (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2))

Next tactic:
refine' ⟨continuous_snd.continuousAt, _⟩
theorem hasMFDerivAt_snd (x : M × M') :
    HasMFDerivAt (I.prod I') I' Prod.snd x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) := by
  refine' ⟨continuous_snd.continuousAt, _⟩
  

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x✝ : M
x : M × M'
⊢ HasFDerivWithinAt (writtenInExtChartAt (ModelWithCorners.prod I I') I' x Prod.snd)
    (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) (range ↑(ModelWithCorners.prod I I'))
    (↑(extChartAt (ModelWithCorners.prod I I') x) x)

Next tactic:
have :
    ∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
      (extChartAt I' x.2 ∘ Prod.snd ∘ (extChartAt (I.prod I') x).symm) y = y.2 := by
    /- porting note: was
    apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
    mfld_set_tac
    -/
    filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy
    rw [extChartAt_prod] at hy
    exact (extChartAt I' x.2).right_inv hy.2
theorem hasMFDerivAt_snd (x : M × M') :
    HasMFDerivAt (I.prod I') I' Prod.snd x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) := by
  refine' ⟨continuous_snd.continuousAt, _⟩
  have :
    ∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
      (extChartAt I' x.2 ∘ Prod.snd ∘ (extChartAt (I.prod I') x).symm) y = y.2 := by
    /- porting note: was
    apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
    mfld_set_tac
    -/
    

Proof state:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x✝ : M
x : M × M'
⊢ ∀ᶠ (y : E × E') in 𝓝[range ↑(ModelWithCorners.prod I I')] ↑(extChartAt (ModelWithCorners.prod I I') x) x,
    (↑(extChartAt I' x.2) ∘ Prod.snd ∘ ↑(PartialEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.2

Next tactic:
filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy
theorem hasMFDerivAt_snd (x : M × M') :
    HasMFDerivAt (I.prod I') I' Prod.snd x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) := by
  refine' ⟨continuous_snd.continuousAt, _⟩
  have :
    ∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
      (extChartAt I' x.2 ∘ Prod.snd ∘ (extChartAt (I.prod I') x).symm) y = y.2 := by
    /- porting note: was
    apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
    mfld_set_tac
    -/
    filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy
    

Proof state:
case h
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x✝ : M
x : M × M'
y : E × E'
hy : y ∈ (extChartAt (ModelWithCorners.prod I I') x).target
⊢ (↑(extChartAt I' x.2) ∘ Prod.snd ∘ ↑(PartialEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.2

Next tactic:
rw [extChartAt_prod] at hy
theorem hasMFDerivAt_snd (x : M × M') :
    HasMFDerivAt (I.prod I') I' Prod.snd x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) := by
  refine' ⟨continuous_snd.continuousAt, _⟩
  have :
    ∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
      (extChartAt I' x.2 ∘ Prod.snd ∘ (extChartAt (I.prod I') x).symm) y = y.2 := by
    /- porting note: was
    apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
    mfld_set_tac
    -/
    filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy
    rw [extChartAt_prod] at hy
    

Proof state:
case h
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
inst✝¹² : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H' M'
inst✝⁶ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁵ : NormedAddCommGroup E''
inst✝⁴ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝³ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝² : TopologicalSpace M''
inst✝¹ : ChartedSpace H'' M''
inst✝ : SmoothManifoldWithCorners I'' M''
s : Set M
x✝ : M
x : M × M'
y : E × E'
hy : y ∈ (PartialEquiv.prod (extChartAt I x.1) (extChartAt I' x.2)).target
⊢ (↑(extChartAt I' x.2) ∘ Prod.snd ∘ ↑(PartialEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.2

Next tactic:
exact (extChartAt I' x.2).right_inv hy.2