ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- reflection_inv
@[simp]
theorem reflection_inv : (reflection K)⁻¹ = reflection K 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K]
Goal: (reflection K)⁻¹ = reflection K
-- reflection_reflection
@[simp]
theorem reflection_reflection (p : E) : reflection K (reflection K p) = p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (p : E)
Goal: (reflection K) ((reflection K) p) = p
-- reflection_involutive
theorem reflection_involutive : Function.Involutive (reflection K) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: Function.Involutive ⇑(reflection K)
-- reflection_trans_reflection
@[simp]
theorem reflection_trans_reflection :
    (reflection K).trans (reflection K) = LinearIsometryEquiv.refl 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: (reflection K).trans (reflection K) = LinearIsometryEquiv.refl 𝕜 E
-- reflection_mul_reflection
@[simp]
theorem reflection_mul_reflection : reflection K * reflection K = 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: reflection K * reflection K = 1
-- reflection_orthogonal_apply
theorem reflection_orthogonal_apply (v : E) : reflection Kᗮ v = -reflection K v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (v : E)
Goal: (reflection Kᗮ) v = -(reflection K) v
-- reflection_orthogonal
theorem reflection_orthogonal : reflection Kᗮ = .trans (reflection K) (.neg _) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: reflection Kᗮ = (reflection K).trans (LinearIsometryEquiv.neg 𝕜)
-- reflection_singleton_apply
theorem reflection_singleton_apply (u v : E) :
    reflection (𝕜 ∙ u) v = 2 • (⟪u, v⟫ / ((‖u‖ : 𝕜) ^ 2)) • u - v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (u v : E)
Goal: (reflection (Submodule.span 𝕜 {u})) v = 2 • (⟪u, v⟫_𝕜 / ↑‖u‖ ^ 2) • u - v
-- reflection_eq_self_iff
theorem reflection_eq_self_iff (x : E) : reflection K x = x ↔ x ∈ K 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] (x : E)
Goal: (reflection K) x = x ↔ x ∈ K
-- reflection_mem_subspace_eq_self
theorem reflection_mem_subspace_eq_self {x : E} (hx : x ∈ K) : reflection K x = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] {x : E} (hx : x ∈ K)
Goal: (reflection K) x = x
-- reflection_map_apply
theorem reflection_map_apply {E E' : Type*} [NormedAddCommGroup E] [NormedAddCommGroup E']
    [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (K : Submodule 𝕜 E)
    [HasOrthogonalProjection K] (x : E') :
    reflection (K.map (f.toLinearEquiv : E →ₗ[𝕜] E')) x = f (reflection K (f.symm x)) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_4} {E' : Type u_5} [NormedAddCommGroup E] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (x : E')
Goal: (reflection (Submodule.map (↑f.toLinearEquiv) K)) x = f ((reflection K) (f.symm x))
-- reflection_map
theorem reflection_map {E E' : Type*} [NormedAddCommGroup E] [NormedAddCommGroup E']
    [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (K : Submodule 𝕜 E)
    [HasOrthogonalProjection K] :
    reflection (K.map (f.toLinearEquiv : E →ₗ[𝕜] E')) = f.symm.trans ((reflection K).trans f) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_4} {E' : Type u_5} [NormedAddCommGroup E] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: reflection (Submodule.map (↑f.toLinearEquiv) K) = f.symm.trans ((reflection K).trans f)
-- reflection_bot
@[simp]
theorem reflection_bot : reflection (⊥ : Submodule 𝕜 E) = LinearIsometryEquiv.neg 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: reflection ⊥ = LinearIsometryEquiv.neg 𝕜
-- Submodule.sup_orthogonal_inf_of_completeSpace
theorem Submodule.sup_orthogonal_inf_of_completeSpace {K₁ K₂ : Submodule 𝕜 E} (h : K₁ ≤ K₂)
    [HasOrthogonalProjection K₁] : K₁ ⊔ K₁ᗮ ⊓ K₂ = K₂ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K₁ K₂ : Submodule 𝕜 E} (h : K₁ ≤ K₂) [HasOrthogonalProjection K₁]
Goal: K₁ ⊔ K₁ᗮ ⊓ K₂ = K₂
-- Submodule.sup_orthogonal_of_completeSpace
theorem Submodule.sup_orthogonal_of_completeSpace [HasOrthogonalProjection K] : K ⊔ Kᗮ = ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K]
Goal: K ⊔ Kᗮ = ⊤
-- Submodule.exists_add_mem_mem_orthogonal
theorem Submodule.exists_add_mem_mem_orthogonal [HasOrthogonalProjection K] (v : E) :
    ∃ y ∈ K, ∃ z ∈ Kᗮ, v = y + z 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (v : E)
Goal: ∃ y ∈ K, ∃ z ∈ Kᗮ, v = y + z
-- Submodule.orthogonal_orthogonal
@[simp]
theorem Submodule.orthogonal_orthogonal [HasOrthogonalProjection K] : Kᗮᗮ = K 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: Kᗮᗮ = K
-- Submodule.orthogonal_orthogonal_eq_closure
theorem Submodule.orthogonal_orthogonal_eq_closure [CompleteSpace E] :
    Kᗮᗮ = K.topologicalClosure 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [CompleteSpace E]
Goal: Kᗮᗮ = K.topologicalClosure
-- Submodule.isCompl_orthogonal_of_completeSpace
theorem Submodule.isCompl_orthogonal_of_completeSpace [HasOrthogonalProjection K] : IsCompl K Kᗮ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K]
Goal: IsCompl K Kᗮ
-- orthogonalComplement_eq_orthogonalComplement
@[simp]
theorem orthogonalComplement_eq_orthogonalComplement {L : Submodule 𝕜 E} [HasOrthogonalProjection K]
    [HasOrthogonalProjection L] : Kᗮ = Lᗮ ↔ K = L 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K L : Submodule 𝕜 E} [HasOrthogonalProjection K] [HasOrthogonalProjection L]
Goal: Kᗮ = Lᗮ ↔ K = L
-- Submodule.orthogonal_eq_bot_iff
@[simp]
theorem Submodule.orthogonal_eq_bot_iff [HasOrthogonalProjection K] : Kᗮ = ⊥ ↔ K = ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K]
Goal: Kᗮ = ⊥ ↔ K = ⊤
-- orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
theorem orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero [HasOrthogonalProjection K]
    {v : E} (hv : v ∈ Kᗮ) : orthogonalProjection K v = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] {v : E} (hv : v ∈ Kᗮ)
Goal: (orthogonalProjection K) v = 0
-- Submodule.IsOrtho.orthogonalProjection_comp_subtypeL
theorem Submodule.IsOrtho.orthogonalProjection_comp_subtypeL {U V : Submodule 𝕜 E}
    [HasOrthogonalProjection U] (h : U ⟂ V) : orthogonalProjection U ∘L V.subtypeL = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {U V : Submodule 𝕜 E} [HasOrthogonalProjection U] (h : U ⟂ V)
Goal: (orthogonalProjection U).comp V.subtypeL = 0
-- orthogonalProjection_comp_subtypeL_eq_zero_iff
theorem orthogonalProjection_comp_subtypeL_eq_zero_iff {U V : Submodule 𝕜 E}
    [HasOrthogonalProjection U] : orthogonalProjection U ∘L V.subtypeL = 0 ↔ U ⟂ V 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {U V : Submodule 𝕜 E} [HasOrthogonalProjection U]
Goal: (orthogonalProjection U).comp V.subtypeL = 0 ↔ U ⟂ V
-- orthogonalProjection_eq_linear_proj
theorem orthogonalProjection_eq_linear_proj [HasOrthogonalProjection K] (x : E) :
    orthogonalProjection K x =
      K.linearProjOfIsCompl _ Submodule.isCompl_orthogonal_of_completeSpace x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] (x : E)
Goal: (orthogonalProjection K) x = (K.linearProjOfIsCompl Kᗮ ⋯) x
-- orthogonalProjection_coe_linearMap_eq_linearProj
theorem orthogonalProjection_coe_linearMap_eq_linearProj [HasOrthogonalProjection K] :
    (orthogonalProjection K : E →ₗ[𝕜] K) =
      K.linearProjOfIsCompl _ Submodule.isCompl_orthogonal_of_completeSpace 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K]
Goal: ↑(orthogonalProjection K) = K.linearProjOfIsCompl Kᗮ ⋯
-- reflection_mem_subspace_orthogonalComplement_eq_neg
theorem reflection_mem_subspace_orthogonalComplement_eq_neg [HasOrthogonalProjection K] {v : E}
    (hv : v ∈ Kᗮ) : reflection K v = -v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] {v : E} (hv : v ∈ Kᗮ)
Goal: (reflection K) v = -v
-- orthogonalProjection_mem_subspace_orthogonal_precomplement_eq_zero
theorem orthogonalProjection_mem_subspace_orthogonal_precomplement_eq_zero
    [HasOrthogonalProjection Kᗮ] {v : E} (hv : v ∈ K) : orthogonalProjection Kᗮ v = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection Kᗮ] {v : E} (hv : v ∈ K)
Goal: (orthogonalProjection Kᗮ) v = 0
-- orthogonalProjection_orthogonalProjection_of_le
theorem orthogonalProjection_orthogonalProjection_of_le {U V : Submodule 𝕜 E}
    [HasOrthogonalProjection U] [HasOrthogonalProjection V] (h : U ≤ V) (x : E) :
    orthogonalProjection U (orthogonalProjection V x) = orthogonalProjection U x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {U V : Submodule 𝕜 E} [HasOrthogonalProjection U] [HasOrthogonalProjection V] (h : U ≤ V) (x : E)
Goal: (orthogonalProjection U) ↑((orthogonalProjection V) x) = (orthogonalProjection U) x
-- orthogonalProjection_tendsto_closure_iSup
theorem orthogonalProjection_tendsto_closure_iSup [CompleteSpace E] {ι : Type*} [SemilatticeSup ι]
    (U : ι → Submodule 𝕜 E) [∀ i, CompleteSpace (U i)] (hU : Monotone U) (x : E) :
    Filter.Tendsto (fun i => (orthogonalProjection (U i) x : E)) atTop
      (𝓝 (orthogonalProjection (⨆ i, U i).topologicalClosure x : E)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {ι : Type u_4} [SemilatticeSup ι] (U : ι → Submodule 𝕜 E) [∀ (i : ι), CompleteSpace ↥(U i)] (hU : Monotone U) (x : E)
Goal: Filter.Tendsto (fun i => ↑((orthogonalProjection (U i)) x)) Filter.atTop
  (𝓝 ↑((orthogonalProjection (⨆ i, U i).topologicalClosure) x))
-- orthogonalProjection_tendsto_self
theorem orthogonalProjection_tendsto_self [CompleteSpace E] {ι : Type*} [SemilatticeSup ι]
    (U : ι → Submodule 𝕜 E) [∀ t, CompleteSpace (U t)] (hU : Monotone U) (x : E)
    (hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure) :
    Filter.Tendsto (fun t => (orthogonalProjection (U t) x : E)) atTop (𝓝 x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {ι : Type u_4} [SemilatticeSup ι] (U : ι → Submodule 𝕜 E) [∀ (t : ι), CompleteSpace ↥(U t)] (hU : Monotone U) (x : E) (hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure)
Goal: Filter.Tendsto (fun t => ↑((orthogonalProjection (U t)) x)) Filter.atTop (𝓝 x)
-- Submodule.triorthogonal_eq_orthogonal
theorem Submodule.triorthogonal_eq_orthogonal [CompleteSpace E] : Kᗮᗮᗮ = Kᗮ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [CompleteSpace E]
Goal: Kᗮᗮᗮ = Kᗮ
-- Submodule.topologicalClosure_eq_top_iff
theorem Submodule.topologicalClosure_eq_top_iff [CompleteSpace E] :
    K.topologicalClosure = ⊤ ↔ Kᗮ = ⊥ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [CompleteSpace E]
Goal: K.topologicalClosure = ⊤ ↔ Kᗮ = ⊥
-- Dense.eq_zero_of_inner_left
theorem eq_zero_of_inner_left (hK : Dense (K : Set E)) (h : ∀ v : K, ⟪x, v⟫ = 0) : x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} {x : E} (hK : Dense ↑K) (h : ∀ (v : ↥K), ⟪x, ↑v⟫_𝕜 = 0)
Goal: x = 0
-- Dense.eq_zero_of_mem_orthogonal
theorem eq_zero_of_mem_orthogonal (hK : Dense (K : Set E)) (h : x ∈ Kᗮ) : x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} {x : E} (hK : Dense ↑K) (h : x ∈ Kᗮ)
Goal: x = 0
-- Dense.eq_of_sub_mem_orthogonal
theorem eq_of_sub_mem_orthogonal (hK : Dense (K : Set E)) (h : x - y ∈ Kᗮ) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} {x y : E} (hK : Dense ↑K) (h : x - y ∈ Kᗮ)
Goal: x = y
-- Dense.eq_of_inner_left
theorem eq_of_inner_left (hK : Dense (K : Set E)) (h : ∀ v : K, ⟪x, v⟫ = ⟪y, v⟫) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} {x y : E} (hK : Dense ↑K) (h : ∀ (v : ↥K), ⟪x, ↑v⟫_𝕜 = ⟪y, ↑v⟫_𝕜)
Goal: x = y
-- Dense.eq_of_inner_right
theorem eq_of_inner_right (hK : Dense (K : Set E)) (h : ∀ v : K, ⟪(v : E), x⟫ = ⟪(v : E), y⟫) :
    x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} {x y : E} (hK : Dense ↑K) (h : ∀ (v : ↥K), ⟪↑v, x⟫_𝕜 = ⟪↑v, y⟫_𝕜)
Goal: x = y
-- Dense.eq_zero_of_inner_right
theorem eq_zero_of_inner_right (hK : Dense (K : Set E)) (h : ∀ v : K, ⟪(v : E), x⟫ = 0) : x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} {x : E} (hK : Dense ↑K) (h : ∀ (v : ↥K), ⟪↑v, x⟫_𝕜 = 0)
Goal: x = 0
-- reflection_mem_subspace_orthogonal_precomplement_eq_neg
theorem reflection_mem_subspace_orthogonal_precomplement_eq_neg [HasOrthogonalProjection K] {v : E}
    (hv : v ∈ K) : reflection Kᗮ v = -v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] {v : E} (hv : v ∈ K)
Goal: (reflection Kᗮ) v = -v
-- orthogonalProjection_orthogonalComplement_singleton_eq_zero
theorem orthogonalProjection_orthogonalComplement_singleton_eq_zero (v : E) :
    orthogonalProjection (𝕜 ∙ v)ᗮ v = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (v : E)
Goal: (orthogonalProjection (Submodule.span 𝕜 {v})ᗮ) v = 0
-- reflection_orthogonalComplement_singleton_eq_neg
theorem reflection_orthogonalComplement_singleton_eq_neg (v : E) : reflection (𝕜 ∙ v)ᗮ v = -v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (v : E)
Goal: (reflection (Submodule.span 𝕜 {v})ᗮ) v = -v
-- reflection_sub
theorem reflection_sub {v w : F} (h : ‖v‖ = ‖w‖) : reflection (ℝ ∙ (v - w))ᗮ v = w 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {v w : F} (h : ‖v‖ = ‖w‖)
Goal: (reflection (Submodule.span ℝ {v - w})ᗮ) v = w
-- orthogonalProjection_add_orthogonalProjection_orthogonal
theorem orthogonalProjection_add_orthogonalProjection_orthogonal [HasOrthogonalProjection K]
    (w : E) : (orthogonalProjection K w : E) + (orthogonalProjection Kᗮ w : E) = w 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (w : E)
Goal: ↑((orthogonalProjection K) w) + ↑((orthogonalProjection Kᗮ) w) = w
-- norm_sq_eq_add_norm_sq_projection
theorem norm_sq_eq_add_norm_sq_projection (x : E) (S : Submodule 𝕜 E) [HasOrthogonalProjection S] :
    ‖x‖ ^ 2 = ‖orthogonalProjection S x‖ ^ 2 + ‖orthogonalProjection Sᗮ x‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E) (S : Submodule 𝕜 E) [HasOrthogonalProjection S]
Goal: ‖x‖ ^ 2 = ‖(orthogonalProjection S) x‖ ^ 2 + ‖(orthogonalProjection Sᗮ) x‖ ^ 2
-- id_eq_sum_orthogonalProjection_self_orthogonalComplement
theorem id_eq_sum_orthogonalProjection_self_orthogonalComplement [HasOrthogonalProjection K] :
    ContinuousLinearMap.id 𝕜 E =
      K.subtypeL.comp (orthogonalProjection K) + Kᗮ.subtypeL.comp (orthogonalProjection Kᗮ) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: ContinuousLinearMap.id 𝕜 E = K.subtypeL.comp (orthogonalProjection K) + Kᗮ.subtypeL.comp (orthogonalProjection Kᗮ)
-- inner_orthogonalProjection_eq_of_mem_right
@[simp high]
theorem inner_orthogonalProjection_eq_of_mem_right [HasOrthogonalProjection K] (u : K) (v : E) :
    ⟪orthogonalProjection K v, u⟫ = ⟪v, u⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (u : ↥K) (v : E)
Goal: ⟪(orthogonalProjection K) v, u⟫_𝕜 = ⟪v, ↑u⟫_𝕜
-- inner_orthogonalProjection_eq_of_mem_left
@[simp high]
theorem inner_orthogonalProjection_eq_of_mem_left [HasOrthogonalProjection K] (u : K) (v : E) :
    ⟪u, orthogonalProjection K v⟫ = ⟪(u : E), v⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (u : ↥K) (v : E)
Goal: ⟪u, (orthogonalProjection K) v⟫_𝕜 = ⟪↑u, v⟫_𝕜
-- inner_orthogonalProjection_left_eq_right
theorem inner_orthogonalProjection_left_eq_right [HasOrthogonalProjection K] (u v : E) :
    ⟪↑(orthogonalProjection K u), v⟫ = ⟪u, orthogonalProjection K v⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K] (u v : E)
Goal: ⟪↑((orthogonalProjection K) u), v⟫_𝕜 = ⟪u, ↑((orthogonalProjection K) v)⟫_𝕜
-- orthogonalProjection_isSymmetric
theorem orthogonalProjection_isSymmetric [HasOrthogonalProjection K] :
    (K.subtypeL ∘L orthogonalProjection K : E →ₗ[𝕜] E).IsSymmetric 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) [HasOrthogonalProjection K]
Goal: (↑(K.subtypeL.comp (orthogonalProjection K))).IsSymmetric
-- Submodule.finrank_add_inf_finrank_orthogonal
theorem Submodule.finrank_add_inf_finrank_orthogonal {K₁ K₂ : Submodule 𝕜 E}
    [FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) :
    finrank 𝕜 K₁ + finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = finrank 𝕜 K₂ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K₁ K₂ : Submodule 𝕜 E} [FiniteDimensional 𝕜 ↥K₂] (h : K₁ ≤ K₂)
Goal: FiniteDimensional.finrank 𝕜 ↥K₁ + FiniteDimensional.finrank 𝕜 ↥(K₁ᗮ ⊓ K₂) = FiniteDimensional.finrank 𝕜 ↥K₂
-- Submodule.finrank_add_inf_finrank_orthogonal'
theorem Submodule.finrank_add_inf_finrank_orthogonal' {K₁ K₂ : Submodule 𝕜 E}
    [FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) {n : ℕ} (h_dim : finrank 𝕜 K₁ + n = finrank 𝕜 K₂) :
    finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K₁ K₂ : Submodule 𝕜 E} [FiniteDimensional 𝕜 ↥K₂] (h : K₁ ≤ K₂) {n : ℕ} (h_dim : FiniteDimensional.finrank 𝕜 ↥K₁ + n = FiniteDimensional.finrank 𝕜 ↥K₂)
Goal: FiniteDimensional.finrank 𝕜 ↥(K₁ᗮ ⊓ K₂) = n
-- Submodule.finrank_add_finrank_orthogonal
theorem Submodule.finrank_add_finrank_orthogonal [FiniteDimensional 𝕜 E] (K : Submodule 𝕜 E) :
    finrank 𝕜 K + finrank 𝕜 Kᗮ = finrank 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (K : Submodule 𝕜 E)
Goal: FiniteDimensional.finrank 𝕜 ↥K + FiniteDimensional.finrank 𝕜 ↥Kᗮ = FiniteDimensional.finrank 𝕜 E
-- Submodule.finrank_add_finrank_orthogonal'
theorem Submodule.finrank_add_finrank_orthogonal' [FiniteDimensional 𝕜 E] {K : Submodule 𝕜 E}
    {n : ℕ} (h_dim : finrank 𝕜 K + n = finrank 𝕜 E) : finrank 𝕜 Kᗮ = n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {K : Submodule 𝕜 E} {n : ℕ} (h_dim : FiniteDimensional.finrank 𝕜 ↥K + n = FiniteDimensional.finrank 𝕜 E)
Goal: FiniteDimensional.finrank 𝕜 ↥Kᗮ = n
-- finrank_orthogonal_span_singleton
theorem finrank_orthogonal_span_singleton {n : ℕ} [_i : Fact (finrank 𝕜 E = n + 1)] {v : E}
    (hv : v ≠ 0) : finrank 𝕜 (𝕜 ∙ v)ᗮ = n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank 𝕜 E = n + 1)] {v : E} (hv : v ≠ 0)
Goal: FiniteDimensional.finrank 𝕜 ↥(Submodule.span 𝕜 {v})ᗮ = n
-- LinearIsometryEquiv.reflections_generate_dim_aux
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] {n : ℕ} (φ : F ≃ₗᵢ[ℝ] F) (hn :
  FiniteDimensional.finrank ℝ
      ↥(LinearMap.ker
            (ContinuousLinearMap.id ℝ F -
              ↑{ toLinearEquiv := φ.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }))ᗮ ≤
    n)
Goal: ∃ l, l.length ≤ n ∧ φ = (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l).prod
-- LinearIsometryEquiv.reflections_generate_dim
theorem LinearIsometryEquiv.reflections_generate_dim [FiniteDimensional ℝ F] (φ : F ≃ₗᵢ[ℝ] F) :
    ∃ l : List F, l.length ≤ finrank ℝ F ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] (φ : F ≃ₗᵢ[ℝ] F)
Goal: ∃ l, l.length ≤ FiniteDimensional.finrank ℝ F ∧ φ = (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l).prod
-- LinearIsometryEquiv.reflections_generate
theorem LinearIsometryEquiv.reflections_generate [FiniteDimensional ℝ F] :
    Subgroup.closure (Set.range fun v : F => reflection (ℝ ∙ v)ᗮ) = ⊤ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F]
Goal: Subgroup.closure (Set.range fun v => reflection (Submodule.span ℝ {v})ᗮ) = ⊤
-- OrthogonalFamily.isInternal_iff_of_isComplete
theorem OrthogonalFamily.isInternal_iff_of_isComplete [DecidableEq ι] {V : ι → Submodule 𝕜 E}
    (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ)
    (hc : IsComplete (↑(iSup V) : Set E)) : DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) (hc : IsComplete ↑(iSup V))
Goal: DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥
-- OrthogonalFamily.isInternal_iff
theorem OrthogonalFamily.isInternal_iff [DecidableEq ι] [FiniteDimensional 𝕜 E]
    {V : ι → Submodule 𝕜 E} (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
    DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [DecidableEq ι] [FiniteDimensional 𝕜 E] {V : ι → Submodule 𝕜 E} (hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ)
Goal: DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥