ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrthogonalFamily.inner_sum
theorem OrthogonalFamily.inner_sum (l₁ l₂ : ∀ i, G i) (s : Finset ι) :
    ⟪∑ i ∈ s, V i (l₁ i), ∑ j ∈ s, V j (l₂ j)⟫ = ∑ i ∈ s, ⟪l₁ i, l₂ i⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) (l₁ l₂ : (i : ι) → G i) (s : Finset ι)
Goal: ⟪∑ i ∈ s, (V i) (l₁ i), ∑ j ∈ s, (V j) (l₂ j)⟫_𝕜 = ∑ i ∈ s, ⟪l₁ i, l₂ i⟫_𝕜
-- OrthogonalFamily.norm_sum
theorem OrthogonalFamily.norm_sum (l : ∀ i, G i) (s : Finset ι) :
    ‖∑ i ∈ s, V i (l i)‖ ^ 2 = ∑ i ∈ s, ‖l i‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) (l : (i : ι) → G i) (s : Finset ι)
Goal: ‖∑ i ∈ s, (V i) (l i)‖ ^ 2 = ∑ i ∈ s, ‖l i‖ ^ 2
-- OrthogonalFamily.comp
theorem OrthogonalFamily.comp {γ : Type*} {f : γ → ι} (hf : Function.Injective f) :
    OrthogonalFamily 𝕜 (fun g => G (f g)) fun g => V (f g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) {γ : Type u_6} {f : γ → ι} (hf : Function.Injective f)
Goal: OrthogonalFamily 𝕜 (fun g => G (f g)) fun g => V (f g)
-- OrthogonalFamily.orthonormal_sigma_orthonormal
theorem OrthogonalFamily.orthonormal_sigma_orthonormal {α : ι → Type*} {v_family : ∀ i, α i → G i}
    (hv_family : ∀ i, Orthonormal 𝕜 (v_family i)) :
    Orthonormal 𝕜 fun a : Σi, α i => V a.1 (v_family a.1 a.2) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) {α : ι → Type u_6} {v_family : (i : ι) → α i → G i} (hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i))
Goal: Orthonormal 𝕜 fun a => (V a.fst) (v_family a.fst a.snd)
-- OrthogonalFamily.norm_sq_diff_sum
theorem OrthogonalFamily.norm_sq_diff_sum [DecidableEq ι] (f : ∀ i, G i) (s₁ s₂ : Finset ι) :
    ‖(∑ i ∈ s₁, V i (f i)) - ∑ i ∈ s₂, V i (f i)‖ ^ 2 =
      (∑ i ∈ s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i ∈ s₂ \ s₁, ‖f i‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) [DecidableEq ι] (f : (i : ι) → G i) (s₁ s₂ : Finset ι)
Goal: ‖∑ i ∈ s₁, (V i) (f i) - ∑ i ∈ s₂, (V i) (f i)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖f i‖ ^ 2
-- OrthogonalFamily.summable_iff_norm_sq_summable
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) [CompleteSpace E] (f : (i : ι) → G i)
Goal: (Summable fun i => (V i) (f i)) ↔ Summable fun i => ‖f i‖ ^ 2
-- OrthogonalFamily.independent
theorem OrthogonalFamily.independent {V : ι → Submodule 𝕜 E}
    (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
    CompleteLattice.Independent V 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {V : ι → Submodule 𝕜 E} (hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ)
Goal: CompleteLattice.Independent V
-- DirectSum.IsInternal.collectedBasis_orthonormal
theorem DirectSum.IsInternal.collectedBasis_orthonormal [DecidableEq ι] {V : ι → Submodule 𝕜 E}
    (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ)
    (hV_sum : DirectSum.IsInternal fun i => V i) {α : ι → Type*}
    {v_family : ∀ i, Basis (α i) 𝕜 (V i)} (hv_family : ∀ i, Orthonormal 𝕜 (v_family i)) :
    Orthonormal 𝕜 (hV_sum.collectedBasis v_family) 
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) (hV_sum : DirectSum.IsInternal fun i => V i) {α : ι → Type u_6} {v_family : (i : ι) → Basis (α i) 𝕜 ↥(V i)} (hv_family : ∀ (i : ι), Orthonormal 𝕜 ⇑(v_family i))
Goal: Orthonormal 𝕜 ⇑(hV_sum.collectedBasis v_family)
-- real_inner_eq_re_inner
theorem real_inner_eq_re_inner (x y : E) :
    @Inner.inner ℝ E (Inner.rclikeToReal 𝕜 E) x y = re ⟪x, y⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ⟪x, y⟫_ℝ = RCLike.re ⟪x, y⟫_𝕜
-- real_inner_I_smul_self
theorem real_inner_I_smul_self (x : E) :
    @Inner.inner ℝ E (Inner.rclikeToReal 𝕜 E) x ((I : 𝕜) • x) = 0 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ⟪x, RCLike.I • x⟫_ℝ = 0
-- Complex.inner
@[simp]
protected theorem Complex.inner (w z : ℂ) : ⟪w, z⟫_ℝ = (conj w * z).re 
Arguments: (w z : ℂ)
Goal: ⟪w, z⟫_ℝ = ((starRingEnd ℂ) w * z).re
-- inner_map_complex
theorem inner_map_complex [NormedAddCommGroup G] [InnerProductSpace ℝ G] (f : G ≃ₗᵢ[ℝ] ℂ)
    (x y : G) : ⟪x, y⟫_ℝ = (conj (f x) * f y).re 
Arguments: {G : Type u_4} [NormedAddCommGroup G] [InnerProductSpace ℝ G] (f : G ≃ₗᵢ[ℝ] ℂ) (x y : G)
Goal: ⟪x, y⟫_ℝ = ((starRingEnd ℂ) (f x) * f y).re
-- continuous_inner
theorem continuous_inner : Continuous fun p : E × E => ⟪p.1, p.2⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: Continuous fun p => ⟪p.1, p.2⟫_𝕜
-- Filter.Tendsto.inner
theorem Filter.Tendsto.inner {f g : α → E} {l : Filter α} {x y : E} (hf : Tendsto f l (𝓝 x))
    (hg : Tendsto g l (𝓝 y)) : Tendsto (fun t => ⟪f t, g t⟫) l (𝓝 ⟪x, y⟫) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {α : Type u_4} {f g : α → E} {l : Filter α} {x y : E} (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (fun t => ⟪f t, g t⟫_𝕜) l (𝓝 ⟪x, y⟫_𝕜)
-- ContinuousWithinAt.inner
theorem ContinuousWithinAt.inner (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun t => ⟪f t, g t⟫) s x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {α : Type u_4} [TopologicalSpace α] {f g : α → E} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (fun t => ⟪f t, g t⟫_𝕜) s x
-- ContinuousAt.inner
theorem ContinuousAt.inner (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun t => ⟪f t, g t⟫) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {α : Type u_4} [TopologicalSpace α] {f g : α → E} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (fun t => ⟪f t, g t⟫_𝕜) x
-- ContinuousOn.inner
theorem ContinuousOn.inner (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun t => ⟪f t, g t⟫) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {α : Type u_4} [TopologicalSpace α] {f g : α → E} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun t => ⟪f t, g t⟫_𝕜) s
-- Continuous.inner
@[continuity]
theorem Continuous.inner (hf : Continuous f) (hg : Continuous g) : Continuous fun t => ⟪f t, g t⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {α : Type u_4} [TopologicalSpace α] {f g : α → E} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun t => ⟪f t, g t⟫_𝕜
-- ContinuousLinearMap.reApplyInnerSelf_apply
theorem ContinuousLinearMap.reApplyInnerSelf_apply (T : E →L[𝕜] E) (x : E) :
    T.reApplyInnerSelf x = re ⟪T x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E) (x : E)
Goal: T.reApplyInnerSelf x = RCLike.re ⟪T x, x⟫_𝕜
-- ContinuousLinearMap.reApplyInnerSelf_continuous
theorem ContinuousLinearMap.reApplyInnerSelf_continuous (T : E →L[𝕜] E) :
    Continuous T.reApplyInnerSelf 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E)
Goal: Continuous T.reApplyInnerSelf
-- ContinuousLinearMap.reApplyInnerSelf_smul
theorem ContinuousLinearMap.reApplyInnerSelf_smul (T : E →L[𝕜] E) (x : E) {c : 𝕜} :
    T.reApplyInnerSelf (c • x) = ‖c‖ ^ 2 * T.reApplyInnerSelf x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E) (x : E) {c : 𝕜}
Goal: T.reApplyInnerSelf (c • x) = ‖c‖ ^ 2 * T.reApplyInnerSelf x
-- UniformSpace.Completion.inner_coe
@[simp]
theorem inner_coe (a b : E) : inner (a : Completion E) (b : Completion E) = (inner a b : 𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (a b : E)
Goal: ⟪↑E a, ↑E b⟫_𝕜 = ⟪a, b⟫_𝕜
-- UniformSpace.Completion.continuous_inner
protected theorem continuous_inner :
    Continuous (uncurry inner : Completion E × Completion E → 𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: Continuous (Function.uncurry inner)
-- UniformSpace.Completion.Continuous.inner
protected theorem Continuous.inner {α : Type*} [TopologicalSpace α] {f g : α → Completion E}
    (hf : Continuous f) (hg : Continuous g) : Continuous (fun x : α => inner (f x) (g x) : α → 𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {α : Type u_4} [TopologicalSpace α] {f g : α → UniformSpace.Completion E} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => ⟪f x, g x⟫_𝕜
-- exists_norm_eq_iInf_of_complete_convex
theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
    (h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K) (h₂ : Convex ℝ K) (u : F)
Goal: ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖
-- norm_eq_iInf_iff_real_inner_le_zero
theorem norm_eq_iInf_iff_real_inner_le_zero {K : Set F} (h : Convex ℝ K) {u : F} {v : F}
    (hv : v ∈ K) : (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {K : Set F} (h : Convex ℝ K) {u v : F} (hv : v ∈ K)
Goal: ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
-- exists_norm_eq_iInf_of_complete_subspace
theorem exists_norm_eq_iInf_of_complete_subspace (h : IsComplete (↑K : Set E)) :
    ∀ u : E, ∃ v ∈ K, ‖u - v‖ = ⨅ w : (K : Set E), ‖u - w‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) (h : IsComplete ↑K) (u : E)
Goal: ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖
-- norm_eq_iInf_iff_real_inner_eq_zero
theorem norm_eq_iInf_iff_real_inner_eq_zero (K : Submodule ℝ F) {u : F} {v : F} (hv : v ∈ K) :
    (‖u - v‖ = ⨅ w : (↑K : Set F), ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (K : Submodule ℝ F) {u v : F} (hv : v ∈ K)
Goal: ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0
-- norm_eq_iInf_iff_inner_eq_zero
theorem norm_eq_iInf_iff_inner_eq_zero {u : E} {v : E} (hv : v ∈ K) :
    (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (K : Submodule 𝕜 E) {u v : E} (hv : v ∈ K)
Goal: ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0
-- HasOrthogonalProjection.exists_orthogonal
class HasOrthogonalProjection (K : Submodule 𝕜 E) : Prop where
  exists_orthogonal (v : E) : ∃ w ∈ K, v - w ∈ Kᗮ
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [self : HasOrthogonalProjection K] (v : E)
Goal: ∃ w ∈ K, v - w ∈ Kᗮ
-- orthogonalProjectionFn_mem
theorem orthogonalProjectionFn_mem (v : E) : orthogonalProjectionFn K v ∈ K 
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: orthogonalProjectionFn K v ∈ K
-- orthogonalProjectionFn_inner_eq_zero
theorem orthogonalProjectionFn_inner_eq_zero (v : E) :
    ∀ w ∈ K, ⟪v - orthogonalProjectionFn K v, w⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] (v w : E)
Goal: w ∈ K → ⟪v - orthogonalProjectionFn K v, w⟫_𝕜 = 0
-- eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero
theorem eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero {u v : E} (hvm : v ∈ K)
    (hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0) : orthogonalProjectionFn K u = 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} (hvm : v ∈ K) (hvo : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0)
Goal: orthogonalProjectionFn K u = v
-- orthogonalProjectionFn_norm_sq
theorem orthogonalProjectionFn_norm_sq (v : E) :
    ‖v‖ * ‖v‖ =
      ‖v - orthogonalProjectionFn K v‖ * ‖v - orthogonalProjectionFn K v‖ +
        ‖orthogonalProjectionFn K v‖ * ‖orthogonalProjectionFn 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: ‖v‖ * ‖v‖ =
  ‖v - orthogonalProjectionFn K v‖ * ‖v - orthogonalProjectionFn K v‖ +
    ‖orthogonalProjectionFn K v‖ * ‖orthogonalProjectionFn K v‖
-- orthogonalProjectionFn_eq
@[simp]
theorem orthogonalProjectionFn_eq (v : E) :
    orthogonalProjectionFn K v = (orthogonalProjection K v : E) 
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: orthogonalProjectionFn K v = ↑((orthogonalProjection K) v)
-- orthogonalProjection_inner_eq_zero
@[simp]
theorem orthogonalProjection_inner_eq_zero (v : E) :
    ∀ w ∈ K, ⟪v - orthogonalProjection K v, w⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] (v w : E)
Goal: w ∈ K → ⟪v - ↑((orthogonalProjection K) v), w⟫_𝕜 = 0
-- sub_orthogonalProjection_mem_orthogonal
@[simp]
theorem sub_orthogonalProjection_mem_orthogonal (v : E) : v - orthogonalProjection K v ∈ Kᗮ 
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: v - ↑((orthogonalProjection K) v) ∈ Kᗮ
-- eq_orthogonalProjection_of_mem_of_inner_eq_zero
theorem eq_orthogonalProjection_of_mem_of_inner_eq_zero {u v : E} (hvm : v ∈ K)
    (hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0) : (orthogonalProjection K 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 v : E} (hvm : v ∈ K) (hvo : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0)
Goal: ↑((orthogonalProjection K) u) = v
-- eq_orthogonalProjection_of_mem_orthogonal
theorem eq_orthogonalProjection_of_mem_orthogonal {u v : E} (hv : v ∈ K)
    (hvo : u - v ∈ Kᗮ) : (orthogonalProjection K 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 v : E} (hv : v ∈ K) (hvo : u - v ∈ Kᗮ)
Goal: ↑((orthogonalProjection K) u) = v
-- eq_orthogonalProjection_of_mem_orthogonal'
theorem eq_orthogonalProjection_of_mem_orthogonal' {u v z : E}
    (hv : v ∈ K) (hz : z ∈ Kᗮ) (hu : u = v + z) : (orthogonalProjection K 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 v z : E} (hv : v ∈ K) (hz : z ∈ Kᗮ) (hu : u = v + z)
Goal: ↑((orthogonalProjection K) u) = v
-- orthogonalProjection_orthogonal_val
@[simp]
theorem orthogonalProjection_orthogonal_val (u : E) :
    (orthogonalProjection Kᗮ u : E) = u - orthogonalProjection K u 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] (u : E)
Goal: ↑((orthogonalProjection Kᗮ) u) = u - ↑((orthogonalProjection K) u)
-- orthogonalProjection_orthogonal
theorem orthogonalProjection_orthogonal (u : E) :
    orthogonalProjection Kᗮ u =
      ⟨u - orthogonalProjection K u, sub_orthogonalProjection_mem_orthogonal _⟩ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] (u : E)
Goal: (orthogonalProjection Kᗮ) u = ⟨u - ↑((orthogonalProjection K) u), ⋯⟩
-- orthogonalProjection_minimal
theorem orthogonalProjection_minimal {U : Submodule 𝕜 E} [HasOrthogonalProjection U] (y : E) :
    ‖y - orthogonalProjection U y‖ = ⨅ x : U, ‖y - x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {U : Submodule 𝕜 E} [HasOrthogonalProjection U] (y : E)
Goal: ‖y - ↑((orthogonalProjection U) y)‖ = ⨅ x, ‖y - ↑x‖
-- eq_orthogonalProjection_of_eq_submodule
theorem eq_orthogonalProjection_of_eq_submodule {K' : Submodule 𝕜 E} [HasOrthogonalProjection K']
    (h : K = K') (u : E) : (orthogonalProjection K u : E) = (orthogonalProjection K' u : E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {K : Submodule 𝕜 E} [HasOrthogonalProjection K] {K' : Submodule 𝕜 E} [HasOrthogonalProjection K'] (h : K = K') (u : E)
Goal: ↑((orthogonalProjection K) u) = ↑((orthogonalProjection K') u)
-- orthogonalProjection_mem_subspace_eq_self
@[simp]
theorem orthogonalProjection_mem_subspace_eq_self (v : K) : orthogonalProjection 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 : ↥K)
Goal: (orthogonalProjection K) ↑v = v
-- orthogonalProjection_eq_self_iff
theorem orthogonalProjection_eq_self_iff {v : E} : (orthogonalProjection K v : E) = v ↔ v ∈ K 
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: ↑((orthogonalProjection K) v) = v ↔ v ∈ K
-- orthogonalProjection_eq_zero_iff
@[simp]
theorem orthogonalProjection_eq_zero_iff {v : E} : orthogonalProjection K v = 0 ↔ v ∈ Kᗮ 
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: (orthogonalProjection K) v = 0 ↔ v ∈ Kᗮ
-- ker_orthogonalProjection
@[simp]
theorem ker_orthogonalProjection : LinearMap.ker (orthogonalProjection 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: LinearMap.ker (orthogonalProjection K) = Kᗮ
-- LinearIsometry.map_orthogonalProjection
theorem LinearIsometry.map_orthogonalProjection {E E' : Type*} [NormedAddCommGroup E]
    [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E →ₗᵢ[𝕜] E')
    (p : Submodule 𝕜 E) [HasOrthogonalProjection p] [HasOrthogonalProjection (p.map f.toLinearMap)]
    (x : E) : f (orthogonalProjection p x) = orthogonalProjection (p.map f.toLinearMap) (f 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') (p : Submodule 𝕜 E) [HasOrthogonalProjection p] [HasOrthogonalProjection (Submodule.map f.toLinearMap p)] (x : E)
Goal: f ↑((orthogonalProjection p) x) = ↑((orthogonalProjection (Submodule.map f.toLinearMap p)) (f x))
-- LinearIsometry.map_orthogonalProjection'
theorem LinearIsometry.map_orthogonalProjection' {E E' : Type*} [NormedAddCommGroup E]
    [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E →ₗᵢ[𝕜] E')
    (p : Submodule 𝕜 E) [HasOrthogonalProjection p] [HasOrthogonalProjection (p.map f)] (x : E) :
    f (orthogonalProjection p x) = orthogonalProjection (p.map f) (f 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') (p : Submodule 𝕜 E) [HasOrthogonalProjection p] [HasOrthogonalProjection (Submodule.map f p)] (x : E)
Goal: f ↑((orthogonalProjection p) x) = ↑((orthogonalProjection (Submodule.map f p)) (f x))
-- orthogonalProjection_map_apply
theorem orthogonalProjection_map_apply {E E' : Type*} [NormedAddCommGroup E]
    [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E')
    (p : Submodule 𝕜 E) [HasOrthogonalProjection p] (x : E') :
    (orthogonalProjection (p.map (f.toLinearEquiv : E →ₗ[𝕜] E')) x : E') =
      f (orthogonalProjection p (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') (p : Submodule 𝕜 E) [HasOrthogonalProjection p] (x : E')
Goal: ↑((orthogonalProjection (Submodule.map (↑f.toLinearEquiv) p)) x) = f ↑((orthogonalProjection p) (f.symm x))
-- orthogonalProjection_bot
@[simp]
theorem orthogonalProjection_bot : orthogonalProjection (⊥ : Submodule 𝕜 E) = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: orthogonalProjection ⊥ = 0
-- orthogonalProjection_norm_le
theorem orthogonalProjection_norm_le : ‖orthogonalProjection 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: ‖orthogonalProjection K‖ ≤ 1
-- smul_orthogonalProjection_singleton
theorem smul_orthogonalProjection_singleton {v : E} (w : E) :
    ((‖v‖ ^ 2 : ℝ) : 𝕜) • (orthogonalProjection (𝕜 ∙ v) w : E) = ⟪v, w⟫ • v 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v : E} (w : E)
Goal: ↑(‖v‖ ^ 2) • ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = ⟪v, w⟫_𝕜 • v
-- orthogonalProjection_singleton
theorem orthogonalProjection_singleton {v : E} (w : E) :
    (orthogonalProjection (𝕜 ∙ v) w : E) = (⟪v, w⟫ / ((‖v‖ ^ 2 : ℝ) : 𝕜)) • v 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v : E} (w : E)
Goal: ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v
-- orthogonalProjection_unit_singleton
theorem orthogonalProjection_unit_singleton {v : E} (hv : ‖v‖ = 1) (w : E) :
    (orthogonalProjection (𝕜 ∙ v) w : E) = ⟪v, w⟫ • v 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v : E} (hv : ‖v‖ = 1) (w : E)
Goal: ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = ⟪v, w⟫_𝕜 • v
-- reflection_apply
theorem reflection_apply (p : E) : reflection K p = 2 • (orthogonalProjection K p : E) - 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) p = 2 • ↑((orthogonalProjection K) p) - p
-- reflection_symm
@[simp]
theorem reflection_symm : (reflection K).symm = 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).symm = reflection K