ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.interior_preimage
theorem interior_preimage (hsurj : Surjective f) (s : Set F) :
    interior (f ⁻¹' s) = f ⁻¹' interior s 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] [CompleteSpace E] (hsurj : Function.Surjective ⇑f) (s : Set F)
Goal: interior (⇑f ⁻¹' s) = ⇑f ⁻¹' interior s
-- ContinuousLinearMap.closure_preimage
theorem closure_preimage (hsurj : Surjective f) (s : Set F) : closure (f ⁻¹' s) = f ⁻¹' closure s 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] [CompleteSpace E] (hsurj : Function.Surjective ⇑f) (s : Set F)
Goal: closure (⇑f ⁻¹' s) = ⇑f ⁻¹' closure s
-- ContinuousLinearMap.frontier_preimage
theorem frontier_preimage (hsurj : Surjective f) (s : Set F) :
    frontier (f ⁻¹' s) = f ⁻¹' frontier s 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] [CompleteSpace E] (hsurj : Function.Surjective ⇑f) (s : Set F)
Goal: frontier (⇑f ⁻¹' s) = ⇑f ⁻¹' frontier s
-- ContinuousLinearMap.exists_nonlinearRightInverse_of_surjective
theorem exists_nonlinearRightInverse_of_surjective (f : E →SL[σ] F)
    (hsurj : LinearMap.range f = ⊤) :
    ∃ fsymm : NonlinearRightInverse f, 0 < fsymm.nnnorm 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hsurj : LinearMap.range f = ⊤)
Goal: ∃ fsymm, 0 < fsymm.nnnorm
-- ContinuousLinearMap.nonlinearRightInverseOfSurjective_def
noncomputable irreducible_def nonlinearRightInverseOfSurjective (f : E →SL[σ] F)
  (hsurj : LinearMap.range f = ⊤) : NonlinearRightInverse f 
Arguments: {𝕜 : Type u_5} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hsurj : LinearMap.range f = ⊤)
Goal: f.nonlinearRightInverseOfSurjective hsurj = Classical.choose ⋯
-- ContinuousLinearMap.nonlinearRightInverseOfSurjective_nnnorm_pos
theorem nonlinearRightInverseOfSurjective_nnnorm_pos (f : E →SL[σ] F)
    (hsurj : LinearMap.range f = ⊤) : 0 < (nonlinearRightInverseOfSurjective f hsurj).nnnorm 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hsurj : LinearMap.range f = ⊤)
Goal: 0 < (f.nonlinearRightInverseOfSurjective hsurj).nnnorm
-- LinearEquiv.continuous_symm
@[continuity]
theorem continuous_symm (e : E ≃ₛₗ[σ] F) (h : Continuous e) : Continuous e.symm 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (e : E ≃ₛₗ[σ] F) (h : Continuous ⇑e)
Goal: Continuous ⇑e.symm
-- LinearEquiv.coeFn_toContinuousLinearEquivOfContinuous
@[simp]
theorem coeFn_toContinuousLinearEquivOfContinuous (e : E ≃ₛₗ[σ] F) (h : Continuous e) :
    ⇑(e.toContinuousLinearEquivOfContinuous h) = e 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (e : E ≃ₛₗ[σ] F) (h : Continuous ⇑e)
Goal: ⇑(e.toContinuousLinearEquivOfContinuous h) = ⇑e
-- LinearEquiv.coeFn_toContinuousLinearEquivOfContinuous_symm
@[simp]
theorem coeFn_toContinuousLinearEquivOfContinuous_symm (e : E ≃ₛₗ[σ] F) (h : Continuous e) :
    ⇑(e.toContinuousLinearEquivOfContinuous h).symm = e.symm 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (e : E ≃ₛₗ[σ] F) (h : Continuous ⇑e)
Goal: ⇑(e.toContinuousLinearEquivOfContinuous h).symm = ⇑e.symm
-- ContinuousLinearMap.coe_linearMap_equivRange
@[simp]
theorem coe_linearMap_equivRange (f : E →SL[σ] F) (hinj : Injective f) (hclo : IsClosed (range f)) :
    f.equivRange hinj hclo = f.rangeRestrict 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hinj : Function.Injective ⇑f) (hclo : IsClosed (Set.range ⇑f))
Goal: ↑(f.equivRange hinj hclo) = f.rangeRestrict
-- ContinuousLinearMap.coe_equivRange
@[simp]
theorem coe_equivRange (f : E →SL[σ] F) (hinj : Injective f) (hclo : IsClosed (range f)) :
    (f.equivRange hinj hclo : E → LinearMap.range f) = f.rangeRestrict 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hinj : Function.Injective ⇑f) (hclo : IsClosed (Set.range ⇑f))
Goal: ⇑(f.equivRange hinj hclo) = ⇑f.rangeRestrict
-- ContinuousLinearEquiv.coeFn_ofBijective
@[simp]
theorem coeFn_ofBijective (f : E →SL[σ] F) (hinj : ker f = ⊥) (hsurj : LinearMap.range f = ⊤) :
    ⇑(ofBijective f hinj hsurj) = f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hinj : LinearMap.ker f = ⊥) (hsurj : LinearMap.range f = ⊤)
Goal: ⇑(ContinuousLinearEquiv.ofBijective f hinj hsurj) = ⇑f
-- ContinuousLinearEquiv.coe_ofBijective
theorem coe_ofBijective (f : E →SL[σ] F) (hinj : ker f = ⊥) (hsurj : LinearMap.range f = ⊤) :
    ↑(ofBijective f hinj hsurj) = f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hinj : LinearMap.ker f = ⊥) (hsurj : LinearMap.range f = ⊤)
Goal: ↑(ContinuousLinearEquiv.ofBijective f hinj hsurj) = f
-- ContinuousLinearEquiv.ofBijective_symm_apply_apply
@[simp]
theorem ofBijective_symm_apply_apply (f : E →SL[σ] F) (hinj : ker f = ⊥)
    (hsurj : LinearMap.range f = ⊤) (x : E) : (ofBijective f hinj hsurj).symm (f x) = x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hinj : LinearMap.ker f = ⊥) (hsurj : LinearMap.range f = ⊤) (x : E)
Goal: (ContinuousLinearEquiv.ofBijective f hinj hsurj).symm (f x) = x
-- ContinuousLinearEquiv.ofBijective_apply_symm_apply
@[simp]
theorem ofBijective_apply_symm_apply (f : E →SL[σ] F) (hinj : ker f = ⊥)
    (hsurj : LinearMap.range f = ⊤) (y : F) : f ((ofBijective f hinj hsurj).symm y) = y 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F) (hinj : LinearMap.ker f = ⊥) (hsurj : LinearMap.range f = ⊤) (y : F)
Goal: f ((ContinuousLinearEquiv.ofBijective f hinj hsurj).symm y) = y
-- ContinuousLinearMap.isUnit_iff_bijective
lemma _root_.ContinuousLinearMap.isUnit_iff_bijective {f : E →L[𝕜] E} :
    IsUnit f ↔ Bijective f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {f : E →L[𝕜] E}
Goal: IsUnit f ↔ Function.Bijective ⇑f
-- ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl
theorem range_eq_map_coprodSubtypeLEquivOfIsCompl {F : Type*} [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (f : E →L[𝕜] F) {G : Submodule 𝕜 F}
    (h : IsCompl (LinearMap.range f) G) [CompleteSpace G] (hker : ker f = ⊥) :
    LinearMap.range f =
      ((⊤ : Submodule 𝕜 E).prod (⊥ : Submodule 𝕜 G)).map
        (f.coprodSubtypeLEquivOfIsCompl h hker : E × G →ₗ[𝕜] F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (f : E →L[𝕜] F) {G : Submodule 𝕜 F} (h : IsCompl (LinearMap.range f) G) [CompleteSpace ↥G] (hker : LinearMap.ker f = ⊥)
Goal: LinearMap.range f = Submodule.map (↑↑(f.coprodSubtypeLEquivOfIsCompl h hker)) (⊤.prod ⊥)
-- ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot
theorem closed_complemented_range_of_isCompl_of_ker_eq_bot {F : Type*} [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (f : E →L[𝕜] F) (G : Submodule 𝕜 F)
    (h : IsCompl (LinearMap.range f) G) (hG : IsClosed (G : Set F)) (hker : ker f = ⊥) :
    IsClosed (LinearMap.range f : Set F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (f : E →L[𝕜] F) (G : Submodule 𝕜 F) (h : IsCompl (LinearMap.range f) G) (hG : IsClosed ↑G) (hker : LinearMap.ker f = ⊥)
Goal: IsClosed ↑(LinearMap.range f)
-- LinearMap.continuous_of_isClosed_graph
theorem LinearMap.continuous_of_isClosed_graph (hg : IsClosed (g.graph : Set <| E × F)) :
    Continuous g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (g : E →ₗ[𝕜] F) (hg : IsClosed ↑g.graph)
Goal: Continuous ⇑g
-- LinearMap.continuous_of_seq_closed_graph
theorem LinearMap.continuous_of_seq_closed_graph
    (hg : ∀ (u : ℕ → E) (x y), Tendsto u atTop (𝓝 x) → Tendsto (g ∘ u) atTop (𝓝 y) → y = g x) :
    Continuous g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (g : E →ₗ[𝕜] F) (hg :
  ∀ (u : ℕ → E) (x : E) (y : F),
    Filter.Tendsto u Filter.atTop (𝓝 x) → Filter.Tendsto (⇑g ∘ u) Filter.atTop (𝓝 y) → y = g x)
Goal: Continuous ⇑g
-- ContinuousLinearMap.coeFn_ofIsClosedGraph
@[simp]
theorem coeFn_ofIsClosedGraph (hg : IsClosed (g.graph : Set <| E × F)) :
    ⇑(ContinuousLinearMap.ofIsClosedGraph hg) = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {g : E →ₗ[𝕜] F} (hg : IsClosed ↑g.graph)
Goal: ⇑(ContinuousLinearMap.ofIsClosedGraph hg) = ⇑g
-- ContinuousLinearMap.coe_ofIsClosedGraph
theorem coe_ofIsClosedGraph (hg : IsClosed (g.graph : Set <| E × F)) :
    ↑(ContinuousLinearMap.ofIsClosedGraph hg) = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {g : E →ₗ[𝕜] F} (hg : IsClosed ↑g.graph)
Goal: ↑(ContinuousLinearMap.ofIsClosedGraph hg) = g
-- ContinuousLinearMap.coeFn_ofSeqClosedGraph
@[simp]
theorem coeFn_ofSeqClosedGraph
    (hg : ∀ (u : ℕ → E) (x y), Tendsto u atTop (𝓝 x) → Tendsto (g ∘ u) atTop (𝓝 y) → y = g x) :
    ⇑(ContinuousLinearMap.ofSeqClosedGraph hg) = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {g : E →ₗ[𝕜] F} (hg :
  ∀ (u : ℕ → E) (x : E) (y : F),
    Filter.Tendsto u Filter.atTop (𝓝 x) → Filter.Tendsto (⇑g ∘ u) Filter.atTop (𝓝 y) → y = g x)
Goal: ⇑(ContinuousLinearMap.ofSeqClosedGraph hg) = ⇑g
-- ContinuousLinearMap.coe_ofSeqClosedGraph
theorem coe_ofSeqClosedGraph
    (hg : ∀ (u : ℕ → E) (x y), Tendsto u atTop (𝓝 x) → Tendsto (g ∘ u) atTop (𝓝 y) → y = g x) :
    ↑(ContinuousLinearMap.ofSeqClosedGraph hg) = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {g : E →ₗ[𝕜] F} (hg :
  ∀ (u : ℕ → E) (x : E) (y : F),
    Filter.Tendsto u Filter.atTop (𝓝 x) → Filter.Tendsto (⇑g ∘ u) Filter.atTop (𝓝 y) → y = g x)
Goal: ↑(ContinuousLinearMap.ofSeqClosedGraph hg) = g
-- ContinuousLinearMap.closed_range_of_antilipschitz
lemma closed_range_of_antilipschitz {f : E →SL[σ] F} {c : ℝ≥0} (hf : AntilipschitzWith c f) :
    (LinearMap.range f).topologicalClosure = LinearMap.range f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace E] {f : E →SL[σ] F} {c : ℝ≥0} (hf : AntilipschitzWith c ⇑f)
Goal: (LinearMap.range f).topologicalClosure = LinearMap.range f
-- ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz
lemma bijective_iff_dense_range_and_antilipschitz (f : E →SL[σ] F) :
    Bijective f ↔ (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] (f : E →SL[σ] F)
Goal: Function.Bijective ⇑f ↔ (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f
-- AntilipschitzWith.completeSpace_range_clm
lemma _root_.AntilipschitzWith.completeSpace_range_clm {f : E →SL[σ] F} {c : ℝ≥0}
    (hf : AntilipschitzWith c f) : CompleteSpace (LinearMap.range f) 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace F] [CompleteSpace E] {f : E →SL[σ] F} {c : ℝ≥0} (hf : AntilipschitzWith c ⇑f)
Goal: CompleteSpace ↥(LinearMap.range f)
-- hasFDerivAtFilter_iff_hasDerivAtFilter
theorem hasFDerivAtFilter_iff_hasDerivAtFilter {f' : 𝕜 →L[𝕜] F} :
    HasFDerivAtFilter f f' x L ↔ HasDerivAtFilter f (f' 1) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {L : Filter 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasFDerivAtFilter f f' x L ↔ HasDerivAtFilter f (f' 1) x L
-- HasFDerivAtFilter.hasDerivAtFilter
theorem HasFDerivAtFilter.hasDerivAtFilter {f' : 𝕜 →L[𝕜] F} :
    HasFDerivAtFilter f f' x L → HasDerivAtFilter f (f' 1) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {L : Filter 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasFDerivAtFilter f f' x L → HasDerivAtFilter f (f' 1) x L
-- hasFDerivWithinAt_iff_hasDerivWithinAt
theorem hasFDerivWithinAt_iff_hasDerivWithinAt {f' : 𝕜 →L[𝕜] F} :
    HasFDerivWithinAt f f' s x ↔ HasDerivWithinAt f (f' 1) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasFDerivWithinAt f f' s x ↔ HasDerivWithinAt f (f' 1) s x
-- hasDerivWithinAt_iff_hasFDerivWithinAt
theorem hasDerivWithinAt_iff_hasFDerivWithinAt {f' : F} :
    HasDerivWithinAt f f' s x ↔ HasFDerivWithinAt f (smulRight (1 : 𝕜 →L[𝕜] 𝕜) f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {f' : F}
Goal: HasDerivWithinAt f f' s x ↔ HasFDerivWithinAt f (ContinuousLinearMap.smulRight 1 f') s x
-- HasFDerivWithinAt.hasDerivWithinAt
theorem HasFDerivWithinAt.hasDerivWithinAt {f' : 𝕜 →L[𝕜] F} :
    HasFDerivWithinAt f f' s x → HasDerivWithinAt f (f' 1) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasFDerivWithinAt f f' s x → HasDerivWithinAt f (f' 1) s x
-- HasDerivWithinAt.hasFDerivWithinAt
theorem HasDerivWithinAt.hasFDerivWithinAt {f' : F} :
    HasDerivWithinAt f f' s x → HasFDerivWithinAt f (smulRight (1 : 𝕜 →L[𝕜] 𝕜) f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {f' : F}
Goal: HasDerivWithinAt f f' s x → HasFDerivWithinAt f (ContinuousLinearMap.smulRight 1 f') s x
-- hasFDerivAt_iff_hasDerivAt
theorem hasFDerivAt_iff_hasDerivAt {f' : 𝕜 →L[𝕜] F} : HasFDerivAt f f' x ↔ HasDerivAt f (f' 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasFDerivAt f f' x ↔ HasDerivAt f (f' 1) x
-- HasFDerivAt.hasDerivAt
theorem HasFDerivAt.hasDerivAt {f' : 𝕜 →L[𝕜] F} : HasFDerivAt f f' x → HasDerivAt f (f' 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasFDerivAt f f' x → HasDerivAt f (f' 1) x
-- hasStrictFDerivAt_iff_hasStrictDerivAt
theorem hasStrictFDerivAt_iff_hasStrictDerivAt {f' : 𝕜 →L[𝕜] F} :
    HasStrictFDerivAt f f' x ↔ HasStrictDerivAt f (f' 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasStrictFDerivAt f f' x ↔ HasStrictDerivAt f (f' 1) x
-- HasStrictFDerivAt.hasStrictDerivAt
protected theorem HasStrictFDerivAt.hasStrictDerivAt {f' : 𝕜 →L[𝕜] F} :
    HasStrictFDerivAt f f' x → HasStrictDerivAt f (f' 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {f' : 𝕜 →L[𝕜] F}
Goal: HasStrictFDerivAt f f' x → HasStrictDerivAt f (f' 1) x
-- hasStrictDerivAt_iff_hasStrictFDerivAt
theorem hasStrictDerivAt_iff_hasStrictFDerivAt :
    HasStrictDerivAt f f' x ↔ HasStrictFDerivAt f (smulRight (1 : 𝕜 →L[𝕜] 𝕜) f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasStrictDerivAt f f' x ↔ HasStrictFDerivAt f (ContinuousLinearMap.smulRight 1 f') x
-- hasDerivAt_iff_hasFDerivAt
theorem hasDerivAt_iff_hasFDerivAt {f' : F} :
    HasDerivAt f f' x ↔ HasFDerivAt f (smulRight (1 : 𝕜 →L[𝕜] 𝕜) f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {f' : F}
Goal: HasDerivAt f f' x ↔ HasFDerivAt f (ContinuousLinearMap.smulRight 1 f') x
-- derivWithin_zero_of_not_differentiableWithinAt
theorem derivWithin_zero_of_not_differentiableWithinAt (h : ¬DifferentiableWithinAt 𝕜 f s x) :
    derivWithin f s x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : ¬DifferentiableWithinAt 𝕜 f s x)
Goal: derivWithin f s x = 0
-- derivWithin_zero_of_isolated
theorem derivWithin_zero_of_isolated (h : 𝓝[s \ {x}] x = ⊥) : derivWithin f s x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : 𝓝[s \ {x}] x = ⊥)
Goal: derivWithin f s x = 0
-- derivWithin_zero_of_nmem_closure
theorem derivWithin_zero_of_nmem_closure (h : x ∉ closure s) : derivWithin f s x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : x ∉ closure s)
Goal: derivWithin f s x = 0
-- differentiableWithinAt_of_derivWithin_ne_zero
theorem differentiableWithinAt_of_derivWithin_ne_zero (h : derivWithin f s x ≠ 0) :
    DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : derivWithin f s x ≠ 0)
Goal: DifferentiableWithinAt 𝕜 f s x
-- deriv_zero_of_not_differentiableAt
theorem deriv_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : deriv f x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (h : ¬DifferentiableAt 𝕜 f x)
Goal: deriv f x = 0
-- differentiableAt_of_deriv_ne_zero
theorem differentiableAt_of_deriv_ne_zero (h : deriv f x ≠ 0) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (h : deriv f x ≠ 0)
Goal: DifferentiableAt 𝕜 f x
-- UniqueDiffWithinAt.eq_deriv
theorem UniqueDiffWithinAt.eq_deriv (s : Set 𝕜) (H : UniqueDiffWithinAt 𝕜 s x)
    (h : HasDerivWithinAt f f' s x) (h₁ : HasDerivWithinAt f f₁' s x) : f' = f₁' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' f₁' : F} {x : 𝕜} (s : Set 𝕜) (H : UniqueDiffWithinAt 𝕜 s x) (h : HasDerivWithinAt f f' s x) (h₁ : HasDerivWithinAt f f₁' s x)
Goal: f' = f₁'
-- hasDerivAtFilter_iff_isLittleO
theorem hasDerivAtFilter_iff_isLittleO :
    HasDerivAtFilter f f' x L ↔ (fun x' : 𝕜 => f x' - f x - (x' - x) • f') =o[L] fun x' => x' - x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜}
Goal: HasDerivAtFilter f f' x L ↔ (fun x' => f x' - f x - (x' - x) • f') =o[L] fun x' => x' - x