ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.hasBasis_nhds_zero_of_basis
protected theorem hasBasis_nhds_zero_of_basis [TopologicalSpace F] [TopologicalAddGroup F]
    {ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
    (𝓝 (0 : E →SL[σ] F)).HasBasis (fun Si : Set E × ι => Bornology.IsVonNBounded 𝕜₁ Si.1 ∧ p Si.2)
      fun Si => { f : E →SL[σ] F | ∀ x ∈ Si.1, f x ∈ b Si.2 } 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] {σ : 𝕜₁ →+* 𝕜₂} {E : Type u_4} {F : Type u_5} [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup F] {ι : Type u_7} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0).HasBasis p b)
Goal: (𝓝 0).HasBasis (fun Si => Bornology.IsVonNBounded 𝕜₁ Si.1 ∧ p Si.2) fun Si => {f | ∀ x ∈ Si.1, f x ∈ b Si.2}
-- ContinuousLinearMap.hasBasis_nhds_zero
protected theorem hasBasis_nhds_zero [TopologicalSpace F] [TopologicalAddGroup F] :
    (𝓝 (0 : E →SL[σ] F)).HasBasis
      (fun SV : Set E × Set F => Bornology.IsVonNBounded 𝕜₁ SV.1 ∧ SV.2 ∈ (𝓝 0 : Filter F))
      fun SV => { f : E →SL[σ] F | ∀ x ∈ SV.1, f x ∈ SV.2 } 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] {σ : 𝕜₁ →+* 𝕜₂} {E : Type u_4} {F : Type u_5} [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup F]
Goal: (𝓝 0).HasBasis (fun SV => Bornology.IsVonNBounded 𝕜₁ SV.1 ∧ SV.2 ∈ 𝓝 0) fun SV => {f | ∀ x ∈ SV.1, f x ∈ SV.2}
-- ContinuousLinearMap.precomp_toFun
@[simps]
def precomp [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [RingHomSurjective σ]
    [RingHomIsometric σ] (L : E →SL[σ] F) : (F →SL[τ] G) →L[𝕜₃] E →SL[ρ] G 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedField 𝕜₃] {σ : 𝕜₁ →+* 𝕜₂} {τ : 𝕜₂ →+* 𝕜₃} {ρ : 𝕜₁ →+* 𝕜₃} [RingHomCompTriple σ τ ρ] {E : Type u_4} {F : Type u_5} (G : Type u_6) [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [AddCommGroup G] [Module 𝕜₃ G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [RingHomSurjective σ] [RingHomIsometric σ] (L : E →SL[σ] F) (f : F →SL[τ] G)
Goal: (ContinuousLinearMap.precomp G L) f = f.comp L
-- ContinuousLinearMap.precomp_apply
@[simps]
def precomp [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [RingHomSurjective σ]
    [RingHomIsometric σ] (L : E →SL[σ] F) : (F →SL[τ] G) →L[𝕜₃] E →SL[ρ] G 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedField 𝕜₃] {σ : 𝕜₁ →+* 𝕜₂} {τ : 𝕜₂ →+* 𝕜₃} {ρ : 𝕜₁ →+* 𝕜₃} [RingHomCompTriple σ τ ρ] {E : Type u_4} {F : Type u_5} (G : Type u_6) [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [AddCommGroup G] [Module 𝕜₃ G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [RingHomSurjective σ] [RingHomIsometric σ] (L : E →SL[σ] F) (f : F →SL[τ] G)
Goal: (ContinuousLinearMap.precomp G L) f = f.comp L
-- ContinuousLinearMap.postcomp_toFun
@[simps]
def postcomp [TopologicalAddGroup F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G]
    [ContinuousConstSMul 𝕜₂ F] (L : F →SL[τ] G) : (E →SL[σ] F) →SL[τ] E →SL[ρ] G 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedField 𝕜₃] {σ : 𝕜₁ →+* 𝕜₂} {τ : 𝕜₂ →+* 𝕜₃} {ρ : 𝕜₁ →+* 𝕜₃} [RingHomCompTriple σ τ ρ] (E : Type u_4) {F : Type u_5} {G : Type u_6} [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [AddCommGroup G] [Module 𝕜₃ G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [ContinuousConstSMul 𝕜₂ F] (L : F →SL[τ] G) (f : E →SL[σ] F)
Goal: (ContinuousLinearMap.postcomp E L) f = L.comp f
-- ContinuousLinearMap.postcomp_apply
@[simps]
def postcomp [TopologicalAddGroup F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G]
    [ContinuousConstSMul 𝕜₂ F] (L : F →SL[τ] G) : (E →SL[σ] F) →SL[τ] E →SL[ρ] G 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedField 𝕜₃] {σ : 𝕜₁ →+* 𝕜₂} {τ : 𝕜₂ →+* 𝕜₃} {ρ : 𝕜₁ →+* 𝕜₃} [RingHomCompTriple σ τ ρ] (E : Type u_4) {F : Type u_5} {G : Type u_6} [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [AddCommGroup G] [Module 𝕜₃ G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [ContinuousConstSMul 𝕜₂ F] (L : F →SL[τ] G) (f : E →SL[σ] F)
Goal: (ContinuousLinearMap.postcomp E L) f = L.comp f
-- ContinuousLinearMap.toLinearMap₂_apply
@[simp] lemma toLinearMap₂_apply (L : E →L[𝕜] F →L[𝕜] G) (v : E) (w : F) :
    L.toLinearMap₂ v w = L v w 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_2} {F : Type u_3} {G : Type u_4} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [AddCommGroup G] [Module 𝕜 G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (L : E →L[𝕜] F →L[𝕜] G) (v : E) (w : F)
Goal: (L.toLinearMap₂ v) w = (L v) w
-- ContinuousLinearEquiv.arrowCongrSL_symm_apply
@[simps]
def arrowCongrSL (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) :
    (E →SL[σ₁₄] H) ≃SL[σ₄₃] F →SL[σ₂₃] G 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {𝕜₄ : Type u_4} {E : Type u_5} {F : Type u_6} {G : Type u_7} {H : Type u_8} [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NontriviallyNormedField 𝕜₄] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜₃ G] [Module 𝕜₄ H] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalSpace H] [TopologicalAddGroup G] [TopologicalAddGroup H] [ContinuousConstSMul 𝕜₃ G] [ContinuousConstSMul 𝕜₄ H] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {σ₃₄ : 𝕜₃ →+* 𝕜₄} {σ₄₃ : 𝕜₄ →+* 𝕜₃} {σ₂₄ : 𝕜₂ →+* 𝕜₄} {σ₁₄ : 𝕜 →+* 𝕜₄} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₁] (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) (L : F →SL[σ₂₃] G)
Goal: (e₁₂.arrowCongrSL e₄₃).symm L = (↑e₄₃.symm).comp (L.comp ↑e₁₂)
-- ContinuousLinearEquiv.arrowCongrSL_apply
@[simps]
def arrowCongrSL (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) :
    (E →SL[σ₁₄] H) ≃SL[σ₄₃] F →SL[σ₂₃] G 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {𝕜₄ : Type u_4} {E : Type u_5} {F : Type u_6} {G : Type u_7} {H : Type u_8} [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NontriviallyNormedField 𝕜₄] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜₃ G] [Module 𝕜₄ H] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalSpace H] [TopologicalAddGroup G] [TopologicalAddGroup H] [ContinuousConstSMul 𝕜₃ G] [ContinuousConstSMul 𝕜₄ H] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {σ₃₄ : 𝕜₃ →+* 𝕜₄} {σ₄₃ : 𝕜₄ →+* 𝕜₃} {σ₂₄ : 𝕜₂ →+* 𝕜₄} {σ₁₄ : 𝕜 →+* 𝕜₄} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₁] (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) (L : E →SL[σ₁₄] H)
Goal: (e₁₂.arrowCongrSL e₄₃) L = (↑e₄₃).comp (L.comp ↑e₁₂.symm)
-- ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_apply
theorem arrowCongrSL_toLinearEquiv_apply (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G)
    (L : E →SL[σ₁₄] H) : (e₁₂.arrowCongrSL e₄₃).toLinearEquiv L =
      (e₄₃ : H →SL[σ₄₃] G).comp (L.comp (e₁₂.symm : F →SL[σ₂₁] E)) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {𝕜₄ : Type u_4} {E : Type u_5} {F : Type u_6} {G : Type u_7} {H : Type u_8} [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NontriviallyNormedField 𝕜₄] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜₃ G] [Module 𝕜₄ H] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalSpace H] [TopologicalAddGroup G] [TopologicalAddGroup H] [ContinuousConstSMul 𝕜₃ G] [ContinuousConstSMul 𝕜₄ H] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {σ₃₄ : 𝕜₃ →+* 𝕜₄} {σ₄₃ : 𝕜₄ →+* 𝕜₃} {σ₂₄ : 𝕜₂ →+* 𝕜₄} {σ₁₄ : 𝕜 →+* 𝕜₄} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₁] (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) (L : E →SL[σ₁₄] H)
Goal: (e₁₂.arrowCongrSL e₄₃).toLinearEquiv L = (↑e₄₃).comp (L.comp ↑e₁₂.symm)
-- ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_symm_apply
@[simp]
theorem arrowCongrSL_toLinearEquiv_symm_apply (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G)
    (L : F →SL[σ₂₃] G) : (e₁₂.arrowCongrSL e₄₃).toLinearEquiv.symm L =
      (e₄₃.symm : G →SL[σ₃₄] H).comp (L.comp (e₁₂ : E →SL[σ₁₂] F)) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {𝕜₄ : Type u_4} {E : Type u_5} {F : Type u_6} {G : Type u_7} {H : Type u_8} [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NontriviallyNormedField 𝕜₄] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜₃ G] [Module 𝕜₄ H] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalSpace H] [TopologicalAddGroup G] [TopologicalAddGroup H] [ContinuousConstSMul 𝕜₃ G] [ContinuousConstSMul 𝕜₄ H] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {σ₃₄ : 𝕜₃ →+* 𝕜₄} {σ₄₃ : 𝕜₄ →+* 𝕜₃} {σ₂₄ : 𝕜₂ →+* 𝕜₄} {σ₁₄ : 𝕜 →+* 𝕜₄} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₁] (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) (L : F →SL[σ₂₃] G)
Goal: (e₁₂.arrowCongrSL e₄₃).symm L = (↑e₄₃.symm).comp (L.comp ↑e₁₂)
-- ContinuousLinearEquiv.arrowCongr_apply
@[simp] lemma arrowCongr_apply (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) (f : E →L[𝕜] H) (x : F) :
    e₁.arrowCongr e₂ f x = e₂ (f (e₁.symm x)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {H : Type u_5} [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H] [NontriviallyNormedField 𝕜] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [Module 𝕜 H] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalSpace H] [TopologicalAddGroup G] [TopologicalAddGroup H] [ContinuousConstSMul 𝕜 G] [ContinuousConstSMul 𝕜 H] (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) (f : E →L[𝕜] H) (x : F)
Goal: ((e₁.arrowCongr e₂) f) x = e₂ (f (e₁.symm x))
-- ContinuousLinearEquiv.arrowCongr_symm
@[simp] lemma arrowCongr_symm (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) :
    (e₁.arrowCongr e₂).symm = e₁.symm.arrowCongr e₂.symm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {H : Type u_5} [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H] [NontriviallyNormedField 𝕜] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [Module 𝕜 H] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalSpace H] [TopologicalAddGroup G] [TopologicalAddGroup H] [ContinuousConstSMul 𝕜 G] [ContinuousConstSMul 𝕜 H] (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G)
Goal: (e₁.arrowCongr e₂).symm = e₁.symm.arrowCongr e₂.symm
-- Specialization.toEquiv_symm
@[simp] lemma toEquiv_symm : (@toEquiv α).symm = ofEquiv 
Arguments: {α : Type u_1}
Goal: Specialization.toEquiv.symm = Specialization.ofEquiv
-- Specialization.ofEquiv_symm
@[simp] lemma ofEquiv_symm : (@ofEquiv α).symm = toEquiv 
Arguments: {α : Type u_1}
Goal: Specialization.ofEquiv.symm = Specialization.toEquiv
-- Specialization.toEquiv_ofEquiv
@[simp] lemma toEquiv_ofEquiv (a : Specialization α) : toEquiv (ofEquiv a) = a 
Arguments: {α : Type u_1} (a : Specialization α)
Goal: Specialization.toEquiv (Specialization.ofEquiv a) = a
-- Specialization.ofEquiv_toEquiv
@[simp] lemma ofEquiv_toEquiv (a : α) : ofEquiv (toEquiv a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: Specialization.ofEquiv (Specialization.toEquiv a) = a
-- Specialization.toEquiv_inj
@[simp, nolint simpNF] lemma toEquiv_inj {a b : α} : toEquiv a = toEquiv b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: Specialization.toEquiv a = Specialization.toEquiv b ↔ a = b
-- Specialization.ofEquiv_inj
@[simp, nolint simpNF] lemma ofEquiv_inj {a b : Specialization α} : ofEquiv a = ofEquiv b ↔ a = b 
Arguments: {α : Type u_1} {a b : Specialization α}
Goal: Specialization.ofEquiv a = Specialization.ofEquiv b ↔ a = b
-- Specialization.toEquiv_le_toEquiv
@[simp] lemma toEquiv_le_toEquiv {a b : α} : toEquiv a ≤ toEquiv b ↔ b ⤳ a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a b : α}
Goal: Specialization.toEquiv a ≤ Specialization.toEquiv b ↔ b ⤳ a
-- Specialization.ofEquiv_specializes_ofEquiv
@[simp] lemma ofEquiv_specializes_ofEquiv {a b : Specialization α} :
  ofEquiv a ⤳ ofEquiv b ↔ b ≤ a 
Arguments: {α : Type u_1} [TopologicalSpace α] {a b : Specialization α}
Goal: Specialization.ofEquiv a ⤳ Specialization.ofEquiv b ↔ b ≤ a
-- Specialization.isOpen_toEquiv_preimage
@[simp] lemma isOpen_toEquiv_preimage [AlexandrovDiscrete α] {s : Set (Specialization α)} :
  IsOpen (toEquiv ⁻¹' s) ↔ IsUpperSet s 
Arguments: {α : Type u_1} [TopologicalSpace α] [AlexandrovDiscrete α] {s : Set (Specialization α)}
Goal: IsOpen (⇑Specialization.toEquiv ⁻¹' s) ↔ IsUpperSet s
-- Specialization.isUpperSet_ofEquiv_preimage
@[simp] lemma isUpperSet_ofEquiv_preimage [AlexandrovDiscrete α] {s : Set α} :
  IsUpperSet (ofEquiv ⁻¹' s) ↔ IsOpen s 
Arguments: {α : Type u_1} [TopologicalSpace α] [AlexandrovDiscrete α] {s : Set α}
Goal: IsUpperSet (⇑Specialization.ofEquiv ⁻¹' s) ↔ IsOpen s
-- Specialization.map_id
@[simp] lemma map_id : map (ContinuousMap.id α) = OrderHom.id 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: Specialization.map (ContinuousMap.id α) = OrderHom.id
-- Specialization.map_comp
@[simp] lemma map_comp (g : C(β, γ)) (f : C(α, β)) : map (g.comp f) = (map g).comp (map f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β))
Goal: Specialization.map (g.comp f) = (Specialization.map g).comp (Specialization.map f)
-- topToPreord_map
@[simps]
def topToPreord : TopCat ⥤ Preord 
Goal: ∀ {X Y : TopCat} (f : C(↑X, ↑Y)), topToPreord.map f = Specialization.map f
-- topToPreord_obj
@[simps]
def topToPreord : TopCat ⥤ Preord 
Arguments: (X : TopCat)
Goal: topToPreord.obj X = Preord.of (Specialization ↑X)
-- ProperVAdd.isProperMap_vadd_pair
class ProperVAdd (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [AddGroup G]
    [AddAction G X] : Prop where
    isProperMap_vadd_pair : IsProperMap (fun gx ↦ (gx.1 +ᵥ gx.2, gx.2) : G × X → X × X)
Arguments: {G : Type u_1} {X : Type u_2} [TopologicalSpace G] [TopologicalSpace X] [AddGroup G] [AddAction G X] [self : ProperVAdd G X]
Goal: IsProperMap fun gx => (gx.1 +ᵥ gx.2, gx.2)
-- ProperSMul.isProperMap_smul_pair
@[to_additive existing (attr := mk_iff)]
class ProperSMul (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [Group G]
    [MulAction G X] : Prop where
    isProperMap_smul_pair : IsProperMap (fun gx ↦ (gx.1 • gx.2, gx.2) : G × X → X × X)
Arguments: {G : Type u_1} {X : Type u_2} [TopologicalSpace G] [TopologicalSpace X] [Group G] [MulAction G X] [self : ProperSMul G X]
Goal: IsProperMap fun gx => (gx.1 • gx.2, gx.2)
-- properVAdd_iff
@[to_additive existing (attr := mk_iff)]
class ProperSMul (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [Group G]
    [MulAction G X] : Prop where
    isProperMap_smul_pair : IsProperMap (fun gx ↦ (gx.1 • gx.2, gx.2) : G × X → X × X)
Arguments: (G : Type u_1) (X : Type u_2) [TopologicalSpace G] [TopologicalSpace X] [AddGroup G] [AddAction G X]
Goal: ProperVAdd G X ↔ IsProperMap fun gx => (gx.1 +ᵥ gx.2, gx.2)
-- properSMul_iff
@[to_additive existing (attr := mk_iff)]
class ProperSMul (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [Group G]
    [MulAction G X] : Prop where
    isProperMap_smul_pair : IsProperMap (fun gx ↦ (gx.1 • gx.2, gx.2) : G × X → X × X)
Arguments: (G : Type u_1) (X : Type u_2) [TopologicalSpace G] [TopologicalSpace X] [Group G] [MulAction G X]
Goal: ProperSMul G X ↔ IsProperMap fun gx => (gx.1 • gx.2, gx.2)
-- properVAdd_iff_continuousVAdd_ultrafilter_tendsto
@[to_additive "A group acts `G` properly on a topological space `X` if and only if
for all ultrafilters `𝒰` on `X`, if `𝒰` converges to `(x₁, x₂)`
along the map `(g, x) ↦ (g • x, x)`, then there exists `g : G` such that `g • x₂ = x₁`
and `𝒰.fst` converges to `g`."]
theorem properSMul_iff_continuousSMul_ultrafilter_tendsto :
    ProperSMul G X ↔ ContinuousSMul G X ∧
      (∀ 𝒰 : Ultrafilter (G × X), ∀ x₁ x₂ : X,
        Tendsto (fun gx : G × X ↦ (gx.1 • gx.2, gx.2)) 𝒰 (𝓝 (x₁, x₂)) →
      ∃ g : G, g • x₂ = x₁ ∧ Tendsto (Prod.fst : G × X → G) 𝒰 (𝓝 g)) 
Arguments: {G : Type u_1} {X : Type u_2} [AddGroup G] [AddAction G X] [TopologicalSpace G] [TopologicalSpace X]
Goal: ProperVAdd G X ↔
  ContinuousVAdd G X ∧
    ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
      Filter.Tendsto (fun gx => (gx.1 +ᵥ gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) →
        ∃ g, g +ᵥ x₂ = x₁ ∧ Filter.Tendsto Prod.fst (↑𝒰) (𝓝 g)
-- properSMul_iff_continuousSMul_ultrafilter_tendsto
@[to_additive "A group acts `G` properly on a topological space `X` if and only if
for all ultrafilters `𝒰` on `X`, if `𝒰` converges to `(x₁, x₂)`
along the map `(g, x) ↦ (g • x, x)`, then there exists `g : G` such that `g • x₂ = x₁`
and `𝒰.fst` converges to `g`."]
theorem properSMul_iff_continuousSMul_ultrafilter_tendsto :
    ProperSMul G X ↔ ContinuousSMul G X ∧
      (∀ 𝒰 : Ultrafilter (G × X), ∀ x₁ x₂ : X,
        Tendsto (fun gx : G × X ↦ (gx.1 • gx.2, gx.2)) 𝒰 (𝓝 (x₁, x₂)) →
      ∃ g : G, g • x₂ = x₁ ∧ Tendsto (Prod.fst : G × X → G) 𝒰 (𝓝 g)) 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X]
Goal: ProperSMul G X ↔
  ContinuousSMul G X ∧
    ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
      Filter.Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) →
        ∃ g, g • x₂ = x₁ ∧ Filter.Tendsto Prod.fst (↑𝒰) (𝓝 g)
-- properSMul_iff_continuousSMul_ultrafilter_tendsto_t2
theorem properSMul_iff_continuousSMul_ultrafilter_tendsto_t2 [T2Space X] :
    ProperSMul G X ↔ ContinuousSMul G X ∧
      (∀ 𝒰 : Ultrafilter (G × X), ∀ x₁ x₂ : X,
        Tendsto (fun gx : G × X ↦ (gx.1 • gx.2, gx.2)) 𝒰 (𝓝 (x₁, x₂)) →
     ∃ g : G, Tendsto (Prod.fst : G × X → G) 𝒰 (𝓝 g)) 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] [T2Space X]
Goal: ProperSMul G X ↔
  ContinuousSMul G X ∧
    ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
      Filter.Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, Filter.Tendsto Prod.fst (↑𝒰) (𝓝 g)
-- t2Space_quotient_addAction_of_properVAdd
@[to_additive "If `G` acts properly on `X`, then the quotient space is Hausdorff (T2)."]
theorem t2Space_quotient_mulAction_of_properSMul [ProperSMul G X] :
    T2Space (Quotient (MulAction.orbitRel G X)) 
Arguments: {G : Type u_1} {X : Type u_2} [AddGroup G] [AddAction G X] [TopologicalSpace G] [TopologicalSpace X] [ProperVAdd G X]
Goal: T2Space (Quotient (AddAction.orbitRel G X))
-- t2Space_quotient_mulAction_of_properSMul
@[to_additive "If `G` acts properly on `X`, then the quotient space is Hausdorff (T2)."]
theorem t2Space_quotient_mulAction_of_properSMul [ProperSMul G X] :
    T2Space (Quotient (MulAction.orbitRel G X)) 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] [ProperSMul G X]
Goal: T2Space (Quotient (MulAction.orbitRel G X))
-- t2Space_of_properSMul_of_t2Group
@[to_additive "If a T2 group acts properly on a topological space,
then this topological space is T2."]
theorem t2Space_of_properSMul_of_t2Group [h_proper : ProperSMul G X] [T2Space G] : T2Space X 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] [h_proper : ProperSMul G X] [T2Space G]
Goal: T2Space X
-- t2Space_of_properVAdd_of_t2AddGroup
@[to_additive "If a T2 group acts properly on a topological space,
then this topological space is T2."]
theorem t2Space_of_properSMul_of_t2Group [h_proper : ProperSMul G X] [T2Space G] : T2Space X 
Arguments: {G : Type u_1} {X : Type u_2} [AddGroup G] [AddAction G X] [TopologicalSpace G] [TopologicalSpace X] [h_proper : ProperVAdd G X] [T2Space G]
Goal: T2Space X
-- properSMul_of_closedEmbedding
@[to_additive "If two groups `H` and `G` act on a topological space `X` such that `G` acts properly
and there exists a group homomorphims `H → G` which is a closed embedding compatible with the
actions, then `H` also acts properly on `X`."]
theorem properSMul_of_closedEmbedding {H : Type*} [Group H] [MulAction H X] [TopologicalSpace H]
    [ProperSMul G X] (f : H →* G) (f_clemb : ClosedEmbedding f)
    (f_compat : ∀ (h : H) (x : X), f h • x = h • x) : ProperSMul H X 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] {H : Type u_5} [Group H] [MulAction H X] [TopologicalSpace H] [ProperSMul G X] (f : H →* G) (f_clemb : ClosedEmbedding ⇑f) (f_compat : ∀ (h : H) (x : X), f h • x = h • x)
Goal: ProperSMul H X
-- properVAdd_of_closedEmbedding
@[to_additive "If two groups `H` and `G` act on a topological space `X` such that `G` acts properly
and there exists a group homomorphims `H → G` which is a closed embedding compatible with the
actions, then `H` also acts properly on `X`."]
theorem properSMul_of_closedEmbedding {H : Type*} [Group H] [MulAction H X] [TopologicalSpace H]
    [ProperSMul G X] (f : H →* G) (f_clemb : ClosedEmbedding f)
    (f_compat : ∀ (h : H) (x : X), f h • x = h • x) : ProperSMul H X 
Arguments: {G : Type u_1} {X : Type u_2} [AddGroup G] [AddAction G X] [TopologicalSpace G] [TopologicalSpace X] {H : Type u_5} [AddGroup H] [AddAction H X] [TopologicalSpace H] [ProperVAdd G X] (f : H →+ G) (f_clemb : ClosedEmbedding ⇑f) (f_compat : ∀ (h : H) (x : X), f h +ᵥ x = h +ᵥ x)
Goal: ProperVAdd H X
-- properlyDiscontinuousSMul_iff_properSMul
theorem properlyDiscontinuousSMul_iff_properSMul [T2Space X] [DiscreteTopology G]
    [ContinuousConstSMul G X]
    (compactlyGenerated : ∀ s : Set (X × X), IsClosed s ↔ ∀ ⦃K⦄, IsCompact K → IsClosed (s ∩ K)) :
    ProperlyDiscontinuousSMul G X ↔ ProperSMul G X 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] [T2Space X] [DiscreteTopology G] [ContinuousConstSMul G X] (compactlyGenerated : ∀ (s : Set (X × X)), IsClosed s ↔ ∀ ⦃K : Set (X × X)⦄, IsCompact K → IsClosed (s ∩ K))
Goal: ProperlyDiscontinuousSMul G X ↔ ProperSMul G X
-- WeaklyLocallyCompactSpace.properlyDiscontinuousSMul_iff_properSMul
theorem WeaklyLocallyCompactSpace.properlyDiscontinuousSMul_iff_properSMul [T2Space X]
    [WeaklyLocallyCompactSpace X] [DiscreteTopology G] [ContinuousConstSMul G X] :
    ProperlyDiscontinuousSMul G X ↔ ProperSMul G X 
Arguments: {G : Type u_1} {X : Type u_2} [Group G] [MulAction G X] [TopologicalSpace G] [TopologicalSpace X] [T2Space X] [WeaklyLocallyCompactSpace X] [DiscreteTopology G] [ContinuousConstSMul G X]
Goal: ProperlyDiscontinuousSMul G X ↔ ProperSMul G X