ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- UniqueMDiffOn.tangentBundle_proj_preimage
theorem UniqueMDiffOn.tangentBundle_proj_preimage (hs : UniqueMDiffOn I s) :
    UniqueMDiffOn I.tangent (π E (TangentSpace I) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {s : Set M} (hs : UniqueMDiffOn I s)
Goal: UniqueMDiffOn I.tangent (Bundle.TotalSpace.proj ⁻¹' s)
-- unique_topology_of_t2
theorem unique_topology_of_t2 {t : TopologicalSpace 𝕜} (h₁ : @TopologicalAddGroup 𝕜 t _)
    (h₂ : @ContinuousSMul 𝕜 𝕜 _ hnorm.toUniformSpace.toTopologicalSpace t) (h₃ : @T2Space 𝕜 t) :
    t = hnorm.toUniformSpace.toTopologicalSpace 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {t : TopologicalSpace 𝕜} (h₁ : TopologicalAddGroup 𝕜) (h₂ : ContinuousSMul 𝕜 𝕜) (h₃ : T2Space 𝕜)
Goal: t = UniformSpace.toTopologicalSpace
-- LinearMap.continuous_of_isClosed_ker
theorem LinearMap.continuous_of_isClosed_ker (l : E →ₗ[𝕜] 𝕜)
    (hl : IsClosed (LinearMap.ker l : Set E)) :
    Continuous l 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] (l : E →ₗ[𝕜] 𝕜) (hl : IsClosed ↑(LinearMap.ker l))
Goal: Continuous ⇑l
-- LinearMap.continuous_iff_isClosed_ker
theorem LinearMap.continuous_iff_isClosed_ker (l : E →ₗ[𝕜] 𝕜) :
    Continuous l ↔ IsClosed (LinearMap.ker l : Set E) 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] (l : E →ₗ[𝕜] 𝕜)
Goal: Continuous ⇑l ↔ IsClosed ↑(LinearMap.ker l)
-- LinearMap.continuous_of_nonzero_on_open
theorem LinearMap.continuous_of_nonzero_on_open (l : E →ₗ[𝕜] 𝕜) (s : Set E) (hs₁ : IsOpen s)
    (hs₂ : s.Nonempty) (hs₃ : ∀ x ∈ s, l x ≠ 0) : Continuous l 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] (l : E →ₗ[𝕜] 𝕜) (s : Set E) (hs₁ : IsOpen s) (hs₂ : s.Nonempty) (hs₃ : ∀ x ∈ s, l x ≠ 0)
Goal: Continuous ⇑l
-- _private.Mathlib.Topology.Algebra.Module.FiniteDimension.0.continuous_equivFun_basis_aux
private theorem continuous_equivFun_basis_aux [T2Space E] {ι : Type v} [Fintype ι]
    (ξ : Basis ι 𝕜 E) : Continuous ξ.equivFun 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] [T2Space E] {ι : Type v} [Fintype ι] (ξ : Basis ι 𝕜 E)
Goal: Continuous ⇑ξ.equivFun
-- LinearMap.continuous_of_finiteDimensional
theorem LinearMap.continuous_of_finiteDimensional [T2Space E] [FiniteDimensional 𝕜 E]
    (f : E →ₗ[𝕜] F') : Continuous f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F' : Type x} [AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F'] [ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F')
Goal: Continuous ⇑f
-- continuous_equivFun_basis
theorem continuous_equivFun_basis [T2Space E] {ι : Type*} [Finite ι] (ξ : Basis ι 𝕜 E) :
    Continuous ξ.equivFun 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] [T2Space E] {ι : Type u_1} [Finite ι] (ξ : Basis ι 𝕜 E)
Goal: Continuous ⇑ξ.equivFun
-- LinearMap.coe_toContinuousLinearMap'
@[simp]
theorem coe_toContinuousLinearMap' (f : E →ₗ[𝕜] F') : ⇑(LinearMap.toContinuousLinearMap f) = f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F' : Type x} [AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F'] [ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F')
Goal: ⇑(LinearMap.toContinuousLinearMap f) = ⇑f
-- LinearMap.coe_toContinuousLinearMap
@[simp]
theorem coe_toContinuousLinearMap (f : E →ₗ[𝕜] F') :
    ((LinearMap.toContinuousLinearMap f) : E →ₗ[𝕜] F') = f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F' : Type x} [AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F'] [ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F')
Goal: ↑(LinearMap.toContinuousLinearMap f) = f
-- LinearMap.coe_toContinuousLinearMap_symm
@[simp]
theorem coe_toContinuousLinearMap_symm :
    ⇑(toContinuousLinearMap : (E →ₗ[𝕜] F') ≃ₗ[𝕜] E →L[𝕜] F').symm =
      ((↑) : (E →L[𝕜] F') → E →ₗ[𝕜] F') 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F' : Type x} [AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F'] [ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E]
Goal: ⇑LinearMap.toContinuousLinearMap.symm = ContinuousLinearMap.toLinearMap
-- LinearMap.det_toContinuousLinearMap
@[simp]
theorem det_toContinuousLinearMap (f : E →ₗ[𝕜] E) :
    (LinearMap.toContinuousLinearMap f).det = LinearMap.det f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] E)
Goal: (LinearMap.toContinuousLinearMap f).det = LinearMap.det f
-- LinearMap.ker_toContinuousLinearMap
@[simp]
theorem ker_toContinuousLinearMap (f : E →ₗ[𝕜] F') :
    ker (LinearMap.toContinuousLinearMap f) = ker f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F' : Type x} [AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F'] [ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F')
Goal: LinearMap.ker (LinearMap.toContinuousLinearMap f) = LinearMap.ker f
-- LinearMap.range_toContinuousLinearMap
@[simp]
theorem range_toContinuousLinearMap (f : E →ₗ[𝕜] F') :
    range (LinearMap.toContinuousLinearMap f) = range f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F' : Type x} [AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F'] [ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F')
Goal: LinearMap.range (LinearMap.toContinuousLinearMap f) = LinearMap.range f
-- LinearMap.isOpenMap_of_finiteDimensional
theorem isOpenMap_of_finiteDimensional (f : F →ₗ[𝕜] E) (hf : Function.Surjective f) :
    IsOpenMap f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : F →ₗ[𝕜] E) (hf : Function.Surjective ⇑f)
Goal: IsOpenMap ⇑f
-- LinearEquiv.coe_toContinuousLinearEquiv
@[simp]
theorem coe_toContinuousLinearEquiv (e : E ≃ₗ[𝕜] F) : (e.toContinuousLinearEquiv : E →ₗ[𝕜] F) = e 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] (e : E ≃ₗ[𝕜] F)
Goal: ↑↑e.toContinuousLinearEquiv = ↑e
-- LinearEquiv.coe_toContinuousLinearEquiv'
@[simp]
theorem coe_toContinuousLinearEquiv' (e : E ≃ₗ[𝕜] F) : (e.toContinuousLinearEquiv : E → F) = e 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] (e : E ≃ₗ[𝕜] F)
Goal: ⇑e.toContinuousLinearEquiv = ⇑e
-- LinearEquiv.coe_toContinuousLinearEquiv_symm
@[simp]
theorem coe_toContinuousLinearEquiv_symm (e : E ≃ₗ[𝕜] F) :
    (e.toContinuousLinearEquiv.symm : F →ₗ[𝕜] E) = e.symm 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] (e : E ≃ₗ[𝕜] F)
Goal: ↑↑e.toContinuousLinearEquiv.symm = ↑e.symm
-- LinearEquiv.coe_toContinuousLinearEquiv_symm'
@[simp]
theorem coe_toContinuousLinearEquiv_symm' (e : E ≃ₗ[𝕜] F) :
    (e.toContinuousLinearEquiv.symm : F → E) = e.symm 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] (e : E ≃ₗ[𝕜] F)
Goal: ⇑e.toContinuousLinearEquiv.symm = ⇑e.symm
-- LinearEquiv.toLinearEquiv_toContinuousLinearEquiv
@[simp]
theorem toLinearEquiv_toContinuousLinearEquiv (e : E ≃ₗ[𝕜] F) :
    e.toContinuousLinearEquiv.toLinearEquiv = e 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] (e : E ≃ₗ[𝕜] F)
Goal: e.toContinuousLinearEquiv.toLinearEquiv = e
-- LinearEquiv.toLinearEquiv_toContinuousLinearEquiv_symm
theorem toLinearEquiv_toContinuousLinearEquiv_symm (e : E ≃ₗ[𝕜] F) :
    e.toContinuousLinearEquiv.symm.toLinearEquiv = e.symm 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] (e : E ≃ₗ[𝕜] F)
Goal: e.toContinuousLinearEquiv.symm.toLinearEquiv = e.symm
-- FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq
theorem FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq
    (cond : finrank 𝕜 E = finrank 𝕜 F) : Nonempty (E ≃L[𝕜] F) 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (cond : FiniteDimensional.finrank 𝕜 E = FiniteDimensional.finrank 𝕜 F)
Goal: Nonempty (E ≃L[𝕜] F)
-- FiniteDimensional.nonempty_continuousLinearEquiv_iff_finrank_eq
theorem FiniteDimensional.nonempty_continuousLinearEquiv_iff_finrank_eq :
    Nonempty (E ≃L[𝕜] F) ↔ finrank 𝕜 E = finrank 𝕜 F 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F]
Goal: Nonempty (E ≃L[𝕜] F) ↔ FiniteDimensional.finrank 𝕜 E = FiniteDimensional.finrank 𝕜 F
-- Basis.coe_constrL
@[simp] -- Porting note: removed `norm_cast`
theorem coe_constrL (v : Basis ι 𝕜 E) (f : ι → F) : (v.constrL f : E →ₗ[𝕜] F) = v.constr 𝕜 f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] [T2Space E] (v : Basis ι 𝕜 E) (f : ι → F)
Goal: ↑(v.constrL f) = (v.constr 𝕜) f
-- Basis.equivFunL_apply
@[simps! apply]
def equivFunL (v : Basis ι 𝕜 E) : E ≃L[𝕜] ι → 𝕜 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] [T2Space E] (v : Basis ι 𝕜 E)
Goal: ∀ (a : E) (a_1 : ι), v.equivFunL a a_1 = (v.repr a) a_1
-- Basis.equivFunL_symm_apply_repr
@[simp]
lemma equivFunL_symm_apply_repr (v : Basis ι 𝕜 E) (x : E) :
    v.equivFunL.symm (v.repr x) = x 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] [T2Space E] (v : Basis ι 𝕜 E) (x : E)
Goal: v.equivFunL.symm ⇑(v.repr x) = x
-- Basis.constrL_apply
@[simp]
theorem constrL_apply {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) (f : ι → F) (e : E) :
    v.constrL f e = ∑ i, v.equivFun e i • f i 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] [T2Space E] {ι : Type u_2} [Fintype ι] (v : Basis ι 𝕜 E) (f : ι → F) (e : E)
Goal: (v.constrL f) e = ∑ i : ι, v.equivFun e i • f i
-- Basis.constrL_basis
@[simp 1100]
theorem constrL_basis (v : Basis ι 𝕜 E) (f : ι → F) (i : ι) : v.constrL f (v i) = f i 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] [T2Space E] (v : Basis ι 𝕜 E) (f : ι → F) (i : ι)
Goal: (v.constrL f) (v i) = f i
-- ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero
@[simp]
theorem coe_toContinuousLinearEquivOfDetNeZero (f : E →L[𝕜] E) (hf : f.det ≠ 0) :
    (f.toContinuousLinearEquivOfDetNeZero hf : E →L[𝕜] E) = f 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →L[𝕜] E) (hf : f.det ≠ 0)
Goal: ↑(f.toContinuousLinearEquivOfDetNeZero hf) = f
-- ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero_apply
@[simp]
theorem toContinuousLinearEquivOfDetNeZero_apply (f : E →L[𝕜] E) (hf : f.det ≠ 0) (x : E) :
    f.toContinuousLinearEquivOfDetNeZero hf x = f x 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →L[𝕜] E) (hf : f.det ≠ 0) (x : E)
Goal: (f.toContinuousLinearEquivOfDetNeZero hf) x = f x
-- Matrix.toLin_finTwoProd_toContinuousLinearMap
theorem _root_.Matrix.toLin_finTwoProd_toContinuousLinearMap (a b c d : 𝕜) :
    LinearMap.toContinuousLinearMap
      (Matrix.toLin (Basis.finTwoProd 𝕜) (Basis.finTwoProd 𝕜) !![a, b; c, d]) =
      (a • ContinuousLinearMap.fst 𝕜 𝕜 𝕜 + b • ContinuousLinearMap.snd 𝕜 𝕜 𝕜).prod
        (c • ContinuousLinearMap.fst 𝕜 𝕜 𝕜 + d • ContinuousLinearMap.snd 𝕜 𝕜 𝕜) 
Arguments: {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] (a b c d : 𝕜)
Goal: LinearMap.toContinuousLinearMap ((Matrix.toLin (Basis.finTwoProd 𝕜) (Basis.finTwoProd 𝕜)) !![a, b; c, d]) =
  (a • ContinuousLinearMap.fst 𝕜 𝕜 𝕜 + b • ContinuousLinearMap.snd 𝕜 𝕜 𝕜).prod
    (c • ContinuousLinearMap.fst 𝕜 𝕜 𝕜 + d • ContinuousLinearMap.snd 𝕜 𝕜 𝕜)
-- FiniteDimensional.complete
theorem FiniteDimensional.complete : CompleteSpace E 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [UniformSpace E] [T2Space E] [UniformAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] [FiniteDimensional 𝕜 E]
Goal: CompleteSpace E
-- Submodule.complete_of_finiteDimensional
theorem Submodule.complete_of_finiteDimensional (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 s] :
    IsComplete (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [UniformSpace E] [T2Space E] [UniformAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 ↥s]
Goal: IsComplete ↑s
-- Submodule.closed_of_finiteDimensional
theorem Submodule.closed_of_finiteDimensional (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 s] :
    IsClosed (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 ↥s]
Goal: IsClosed ↑s
-- LinearMap.closedEmbedding_of_injective
theorem LinearMap.closedEmbedding_of_injective [FiniteDimensional 𝕜 E] {f : E →ₗ[𝕜] F}
    (hf : LinearMap.ker f = ⊥) : ClosedEmbedding f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] [AddCommGroup F] [TopologicalSpace F] [T2Space F] [TopologicalAddGroup F] [Module 𝕜 F] [ContinuousSMul 𝕜 F] [FiniteDimensional 𝕜 E] {f : E →ₗ[𝕜] F} (hf : LinearMap.ker f = ⊥)
Goal: ClosedEmbedding ⇑f
-- closedEmbedding_smul_left
theorem closedEmbedding_smul_left {c : E} (hc : c ≠ 0) : ClosedEmbedding fun x : 𝕜 => x • c 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] {c : E} (hc : c ≠ 0)
Goal: ClosedEmbedding fun x => x • c
-- isClosedMap_smul_left
theorem isClosedMap_smul_left (c : E) : IsClosedMap fun x : 𝕜 => x • c 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] (c : E)
Goal: IsClosedMap fun x => x • c
-- ContinuousLinearMap.exists_right_inverse_of_surjective
theorem ContinuousLinearMap.exists_right_inverse_of_surjective [FiniteDimensional 𝕜 F]
    (f : E →L[𝕜] F) (hf : LinearMap.range f = ⊤) :
    ∃ g : F →L[𝕜] E, f.comp g = ContinuousLinearMap.id 𝕜 F 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] [AddCommGroup F] [TopologicalSpace F] [T2Space F] [TopologicalAddGroup F] [Module 𝕜 F] [ContinuousSMul 𝕜 F] [FiniteDimensional 𝕜 F] (f : E →L[𝕜] F) (hf : LinearMap.range f = ⊤)
Goal: ∃ g, f.comp g = ContinuousLinearMap.id 𝕜 F
-- differentiable_within_at_localInvariantProp
theorem differentiable_within_at_localInvariantProp :
    (contDiffGroupoid ⊤ I).LocalInvariantProp (contDiffGroupoid ⊤ I')
      (DifferentiableWithinAtProp I I') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H')
Goal: (contDiffGroupoid ⊤ I).LocalInvariantProp (contDiffGroupoid ⊤ I') (DifferentiableWithinAtProp I I')
-- mdifferentiableWithinAt_iff'
theorem mdifferentiableWithinAt_iff' (f : M → M') (s : Set M) (x : M) :
    MDifferentiableWithinAt I I' f s x ↔ ContinuousWithinAt f s x ∧
    DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f)
      ((extChartAt I x).symm ⁻¹' s ∩ range I) ((extChartAt I x) x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (s : Set M) (x : M)
Goal: MDifferentiableWithinAt I I' f s x ↔
  ContinuousWithinAt f s x ∧
    DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I)
      (↑(extChartAt I x) x)
-- MDifferentiableWithinAt.continuousWithinAt
theorem MDifferentiableWithinAt.continuousWithinAt {f : M → M'} {s : Set M} {x : M}
    (hf : MDifferentiableWithinAt I I' f s x) :
    ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} (hf : MDifferentiableWithinAt I I' f s x)
Goal: ContinuousWithinAt f s x
-- MDifferentiableWithinAt.differentiableWithinAt_writtenInExtChartAt
theorem MDifferentiableWithinAt.differentiableWithinAt_writtenInExtChartAt
    {f : M → M'} {s : Set M} {x : M} (hf : MDifferentiableWithinAt I I' f s x) :
    DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f)
      ((extChartAt I x).symm ⁻¹' s ∩ range I) ((extChartAt I x) x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} (hf : MDifferentiableWithinAt I I' f s x)
Goal: DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I)
  (↑(extChartAt I x) x)
-- mdifferentiableAt_iff
theorem mdifferentiableAt_iff (f : M → M') (x : M) :
    MDifferentiableAt I I' f x ↔ ContinuousAt f x ∧
    DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (range I) ((extChartAt I x) x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (x : M)
Goal: MDifferentiableAt I I' f x ↔
  ContinuousAt f x ∧ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (Set.range ↑I) (↑(extChartAt I x) x)