-- Mathlib/Geometry/Manifold/Instances/Sphere.lean
theorem range_mfderiv_coe_sphere {n : ℕ} [Fact (finrank ℝ E = n + 1)] (v : sphere (0 : E) 1) :
    LinearMap.range (mfderiv (𝓡 n) 𝓘(ℝ, E) ((↑) : sphere (0 : E) 1 → E) v :
    TangentSpace (𝓡 n) v →L[ℝ] E) = (ℝ ∙ (v : E))ᗮ

Proof state:
case h.e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ (Submodule.span ℝ {↑v})ᗮ = (Submodule.span ℝ {-↑v})ᗮ

Tactic:
congr 1

Result:
case h.e'_3.e_K
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ Submodule.span ℝ {↑v} = Submodule.span ℝ {-↑v}

Proof state:
case h.e'_3.e_K
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ Submodule.span ℝ {↑v} = Submodule.span ℝ {-↑v}

Tactic:
apply Submodule.span_eq_span

Result:
case h.e'_3.e_K.hs
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ {↑v} ⊆ ↑(Submodule.span ℝ {-↑v})

case h.e'_3.e_K.ht
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ {-↑v} ⊆ ↑(Submodule.span ℝ {↑v})

Proof state:
case h.e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ LinearMap.range ((Submodule.span ℝ {↑(-v)})ᗮ.subtypeL.comp ↑U.symm.toContinuousLinearEquiv) = (Submodule.span ℝ {↑v})ᗮ

Tactic:
convert
  (U.symm : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] (ℝ ∙ (↑(-v) : E))ᗮ).range_comp
    (ℝ ∙ (↑(-v) : E))ᗮ.subtype using 1

Result:
case h.e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ (Submodule.span ℝ {↑v})ᗮ = LinearMap.range (Submodule.span ℝ {↑(-v)})ᗮ.subtype

Proof state:
case h.e'_3.e_K.ht
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ -↑v ∈ Submodule.span ℝ {↑v}

Tactic:
rw [Submodule.neg_mem_iff]

Result:
case h.e'_3.e_K.ht
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ ↑v ∈ Submodule.span ℝ {↑v}

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ LinearMap.range
      (fderivWithin ℝ (Subtype.val ∘ ↑(ChartedSpace.chartAt v).symm) (Set.range id) (↑(ChartedSpace.chartAt v) v)) =
    (Submodule.span ℝ {↑v})ᗮ

Tactic:
simp only [chartAt, stereographic_neg_apply, fderivWithin_univ,
  LinearIsometryEquiv.toHomeomorph_symm, LinearIsometryEquiv.coe_toHomeomorph,
  LinearIsometryEquiv.map_zero, mfld_simps]

Result:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ LinearMap.range (fderiv ℝ (Subtype.val ∘ ↑(ChartedSpace.chartAt v).symm) (↑(ChartedSpace.chartAt v) v)) =
    (Submodule.span ℝ {↑v})ᗮ

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
⊢ LinearMap.range (fderiv ℝ (Subtype.val ∘ ↑(ChartedSpace.chartAt v).symm) (↑(ChartedSpace.chartAt v) v)) =
    (Submodule.span ℝ {↑v})ᗮ

Tactic:
suffices
    LinearMap.range (fderiv ℝ ((stereoInvFunAux (-v : E) ∘ (↑)) ∘ U.symm) 0) = (ℝ ∙ (v : E))ᗮ by
  convert this using 3
  apply stereographic'_neg

Result:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
⊢ LinearMap.range (fderiv ℝ ((stereoInvFunAux (-↑v) ∘ Subtype.val) ∘ ⇑U.symm) 0) = (Submodule.span ℝ {↑v})ᗮ

Proof state:
case h.e'_3.e_K.hs
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ ↑v ∈ Submodule.span ℝ {-↑v}

Tactic:
rw [← Submodule.neg_mem_iff]

Result:
case h.e'_3.e_K.hs
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
  (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (Submodule.span ℝ {↑(-v)})ᗮ.subtypeL (U.symm 0)
⊢ -↑v ∈ Submodule.span ℝ {-↑v}

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ LinearMap.range
      (fderivWithin ℝ (writtenInExtChartAt (𝓡 n) 𝓘(ℝ, E) v Subtype.val) (Set.range ↑(𝓡 n)) (↑(extChartAt (𝓡 n) v) v)) =
    (Submodule.span ℝ {↑v})ᗮ

Tactic:
dsimp [chartAt]

Result:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ LinearMap.range
      (fderivWithin ℝ (Subtype.val ∘ ↑(ChartedSpace.chartAt v).symm) (Set.range id) (↑(ChartedSpace.chartAt v) v)) =
    (Submodule.span ℝ {↑v})ᗮ