ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- WithLp.nnnorm_equiv_symm_snd
@[simp]
theorem nnnorm_equiv_symm_snd (y : β) :
    ‖(WithLp.equiv p (α × β)).symm (0, y)‖₊ = ‖y‖₊ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (y : β)
Goal: ‖(WithLp.equiv p (α × β)).symm (0, y)‖₊ = ‖y‖₊
-- WithLp.norm_equiv_symm_fst
@[simp]
theorem norm_equiv_symm_fst (x : α) : ‖(WithLp.equiv p (α × β)).symm (x, 0)‖ = ‖x‖ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x : α)
Goal: ‖(WithLp.equiv p (α × β)).symm (x, 0)‖ = ‖x‖
-- WithLp.norm_equiv_symm_snd
@[simp]
theorem norm_equiv_symm_snd (y : β) : ‖(WithLp.equiv p (α × β)).symm (0, y)‖ = ‖y‖ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (y : β)
Goal: ‖(WithLp.equiv p (α × β)).symm (0, y)‖ = ‖y‖
-- WithLp.nndist_equiv_symm_fst
@[simp]
theorem nndist_equiv_symm_fst (x₁ x₂ : α) :
    nndist ((WithLp.equiv p (α × β)).symm (x₁, 0)) ((WithLp.equiv p (α × β)).symm (x₂, 0)) =
      nndist x₁ x₂ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x₁ x₂ : α)
Goal: nndist ((WithLp.equiv p (α × β)).symm (x₁, 0)) ((WithLp.equiv p (α × β)).symm (x₂, 0)) = nndist x₁ x₂
-- WithLp.nndist_equiv_symm_snd
@[simp]
theorem nndist_equiv_symm_snd (y₁ y₂ : β) :
    nndist ((WithLp.equiv p (α × β)).symm (0, y₁)) ((WithLp.equiv p (α × β)).symm (0, y₂)) =
      nndist y₁ y₂ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (y₁ y₂ : β)
Goal: nndist ((WithLp.equiv p (α × β)).symm (0, y₁)) ((WithLp.equiv p (α × β)).symm (0, y₂)) = nndist y₁ y₂
-- WithLp.dist_equiv_symm_fst
@[simp]
theorem dist_equiv_symm_fst (x₁ x₂ : α) :
    dist ((WithLp.equiv p (α × β)).symm (x₁, 0)) ((WithLp.equiv p (α × β)).symm (x₂, 0)) =
      dist x₁ x₂ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x₁ x₂ : α)
Goal: dist ((WithLp.equiv p (α × β)).symm (x₁, 0)) ((WithLp.equiv p (α × β)).symm (x₂, 0)) = dist x₁ x₂
-- WithLp.dist_equiv_symm_snd
@[simp]
theorem dist_equiv_symm_snd (y₁ y₂ : β) :
    dist ((WithLp.equiv p (α × β)).symm (0, y₁)) ((WithLp.equiv p (α × β)).symm (0, y₂)) =
      dist y₁ y₂ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (y₁ y₂ : β)
Goal: dist ((WithLp.equiv p (α × β)).symm (0, y₁)) ((WithLp.equiv p (α × β)).symm (0, y₂)) = dist y₁ y₂
-- WithLp.edist_equiv_symm_fst
@[simp]
theorem edist_equiv_symm_fst (x₁ x₂ : α) :
    edist ((WithLp.equiv p (α × β)).symm (x₁, 0)) ((WithLp.equiv p (α × β)).symm (x₂, 0)) =
      edist x₁ x₂ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x₁ x₂ : α)
Goal: edist ((WithLp.equiv p (α × β)).symm (x₁, 0)) ((WithLp.equiv p (α × β)).symm (x₂, 0)) = edist x₁ x₂
-- WithLp.edist_equiv_symm_snd
@[simp]
theorem edist_equiv_symm_snd (y₁ y₂ : β) :
    edist ((WithLp.equiv p (α × β)).symm (0, y₁)) ((WithLp.equiv p (α × β)).symm (0, y₂)) =
      edist y₁ y₂ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (y₁ y₂ : β)
Goal: edist ((WithLp.equiv p (α × β)).symm (0, y₁)) ((WithLp.equiv p (α × β)).symm (0, y₂)) = edist y₁ y₂
-- coe_neg_sphere
@[simp]
theorem coe_neg_sphere {r : ℝ} (v : sphere (0 : E) r) : ↑(-v) = (-v : E) 
Arguments: {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} (v : ↑(Metric.sphere 0 r))
Goal: ↑(-v) = -↑v
-- coe_neg_ball
@[simp] theorem coe_neg_ball {r : ℝ} (v : ball (0 : E) r) : ↑(-v) = (-v : E) 
Arguments: {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} (v : ↑(Metric.ball 0 r))
Goal: ↑(-v) = -↑v
-- coe_neg_closedBall
@[simp] theorem coe_neg_closedBall {r : ℝ} (v : closedBall (0 : E) r) : ↑(-v) = (-v : E) 
Arguments: {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} (v : ↑(Metric.closedBall 0 r))
Goal: ↑(-v) = -↑v
-- ContinuousLinearMap.toWOT_apply
lemma _root_.ContinuousLinearMap.toWOT_apply {A : E →L[𝕜] F} {x : E} :
    ((ContinuousLinearMap.toWOT 𝕜 E F) A) x = A x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {A : E →L[𝕜] F} {x : E}
Goal: ((ContinuousLinearMap.toWOT 𝕜 E F) A) x = A x
-- ContinuousLinearMapWOT.ext
lemma ext {A B : E →WOT[𝕜] F} (h : ∀ x, A x = B x) : A = B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {A B : E →WOT[𝕜]F} (h : ∀ (x : E), A x = B x)
Goal: A = B
-- ContinuousLinearMapWOT.ext_iff
lemma ext_iff {A B : E →WOT[𝕜] F} : A = B ↔ ∀ x, A x = B x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {A B : E →WOT[𝕜]F}
Goal: A = B ↔ ∀ (x : E), A x = B x
-- ContinuousLinearMapWOT.ext_dual
@[ext 900]
lemma ext_dual {A B : E →WOT[𝕜] F} (h : ∀ x (y : F⋆), y (A x) = y (B x)) : A = B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {A B : E →WOT[𝕜]F} (h : ∀ (x : E) (y : NormedSpace.Dual 𝕜 F), y (A x) = y (B x))
Goal: A = B
-- ContinuousLinearMapWOT.ext_dual_iff
lemma ext_dual_iff {A B : E →WOT[𝕜] F} :
    A = B ↔ ∀ x (y : F⋆), y (A x) = y (B x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {A B : E →WOT[𝕜]F}
Goal: A = B ↔ ∀ (x : E) (y : NormedSpace.Dual 𝕜 F), y (A x) = y (B x)
-- ContinuousLinearMapWOT.zero_apply
@[simp] lemma zero_apply (x : E) : (0 : E →WOT[𝕜] F) x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : E)
Goal: 0 x = 0
-- ContinuousLinearMapWOT.add_apply
@[simp] lemma add_apply {f g : E →WOT[𝕜] F} (x : E) : (f + g) x = f x + g x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E →WOT[𝕜]F} (x : E)
Goal: (f + g) x = f x + g x
-- ContinuousLinearMapWOT.sub_apply
@[simp] lemma sub_apply {f g : E →WOT[𝕜] F} (x : E) : (f - g) x = f x - g x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E →WOT[𝕜]F} (x : E)
Goal: (f - g) x = f x - g x
-- ContinuousLinearMapWOT.neg_apply
@[simp] lemma neg_apply {f : E →WOT[𝕜] F} (x : E) : (-f) x = -(f x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E →WOT[𝕜]F} (x : E)
Goal: (-f) x = -f x
-- ContinuousLinearMapWOT.smul_apply
@[simp] lemma smul_apply {f : E →WOT[𝕜] F} (c : 𝕜) (x : E) : (c • f) x = c • (f x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E →WOT[𝕜]F} (c : 𝕜) (x : E)
Goal: (c • f) x = c • f x
-- ContinuousLinearMapWOT.continuous_inducingFn
@[fun_prop]
lemma continuous_inducingFn : Continuous (inducingFn 𝕜 E F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: Continuous ⇑(ContinuousLinearMapWOT.inducingFn 𝕜 E F)
-- ContinuousLinearMapWOT.continuous_dual_apply
lemma continuous_dual_apply (x : E) (y : F⋆) : Continuous fun (A : E →WOT[𝕜] F) => y (A x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : E) (y : NormedSpace.Dual 𝕜 F)
Goal: Continuous fun A => y (A x)
-- ContinuousLinearMapWOT.continuous_of_dual_apply_continuous
@[fun_prop]
lemma continuous_of_dual_apply_continuous {α : Type*} [TopologicalSpace α] {g : α → E →WOT[𝕜] F}
    (h : ∀ x (y : F⋆), Continuous fun a => y (g a x)) : Continuous g 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] {g : α → E →WOT[𝕜]F} (h : ∀ (x : E) (y : NormedSpace.Dual 𝕜 F), Continuous fun a => y ((g a) x))
Goal: Continuous g
-- ContinuousLinearMapWOT.embedding_inducingFn
lemma embedding_inducingFn : Embedding (inducingFn 𝕜 E F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: Embedding ⇑(ContinuousLinearMapWOT.inducingFn 𝕜 E F)
-- ContinuousLinearMapWOT.tendsto_iff_forall_dual_apply_tendsto
lemma tendsto_iff_forall_dual_apply_tendsto {α : Type*} {l : Filter α} {f : α → E →WOT[𝕜] F}
    {A : E →WOT[𝕜] F} :
    Tendsto f l (𝓝 A) ↔ ∀ x (y : F⋆), Tendsto (fun a => y (f a x)) l (𝓝 (y (A x))) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} {l : Filter α} {f : α → E →WOT[𝕜]F} {A : E →WOT[𝕜]F}
Goal: Filter.Tendsto f l (𝓝 A) ↔ ∀ (x : E) (y : NormedSpace.Dual 𝕜 F), Filter.Tendsto (fun a => y ((f a) x)) l (𝓝 (y (A x)))
-- ContinuousLinearMapWOT.le_nhds_iff_forall_dual_apply_le_nhds
lemma le_nhds_iff_forall_dual_apply_le_nhds {l : Filter (E →WOT[𝕜] F)} {A : E →WOT[𝕜] F} :
    l ≤ 𝓝 A ↔ ∀ x (y : F⋆), l.map (fun T => y (T x)) ≤ 𝓝 (y (A x)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {l : Filter (E →WOT[𝕜]F)} {A : E →WOT[𝕜]F}
Goal: l ≤ 𝓝 A ↔ ∀ (x : E) (y : NormedSpace.Dual 𝕜 F), Filter.map (fun T => y (T x)) l ≤ 𝓝 (y (A x))
-- ContinuousLinearMapWOT.hasBasis_seminorms
lemma hasBasis_seminorms : (𝓝 (0 : E →WOT[𝕜] F)).HasBasis (seminormFamily 𝕜 E F).basisSets id 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: (𝓝 0).HasBasis (ContinuousLinearMapWOT.seminormFamily 𝕜 E F).basisSets id
-- ContinuousLinearMapWOT.withSeminorms
lemma withSeminorms : WithSeminorms (seminormFamily 𝕜 E F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: WithSeminorms (ContinuousLinearMapWOT.seminormFamily 𝕜 E F)
-- ContinuousLinearMap.continuous_toWOT
@[continuity, fun_prop]
lemma ContinuousLinearMap.continuous_toWOT :
    Continuous (ContinuousLinearMap.toWOT 𝕜 E F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: Continuous ⇑(ContinuousLinearMap.toWOT 𝕜 E F)
-- isCompactOperator_zero
theorem isCompactOperator_zero {M₁ M₂ : Type*} [Zero M₁] [TopologicalSpace M₁]
    [TopologicalSpace M₂] [Zero M₂] : IsCompactOperator (0 : M₁ → M₂) 
Arguments: {M₁ : Type u_1} {M₂ : Type u_2} [Zero M₁] [TopologicalSpace M₁] [TopologicalSpace M₂] [Zero M₂]
Goal: IsCompactOperator 0
-- isCompactOperator_iff_exists_mem_nhds_image_subset_compact
theorem isCompactOperator_iff_exists_mem_nhds_image_subset_compact (f : M₁ → M₂) :
    IsCompactOperator f ↔ ∃ V ∈ (𝓝 0 : Filter M₁), ∃ K : Set M₂, IsCompact K ∧ f '' V ⊆ K 
Arguments: {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] (f : M₁ → M₂)
Goal: IsCompactOperator f ↔ ∃ V ∈ 𝓝 0, ∃ K, IsCompact K ∧ f '' V ⊆ K
-- isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image
theorem isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image [T2Space M₂] (f : M₁ → M₂) :
    IsCompactOperator f ↔ ∃ V ∈ (𝓝 0 : Filter M₁), IsCompact (closure <| f '' V) 
Arguments: {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [T2Space M₂] (f : M₁ → M₂)
Goal: IsCompactOperator f ↔ ∃ V ∈ 𝓝 0, IsCompact (closure (f '' V))
-- IsCompactOperator.image_subset_compact_of_isVonNBounded
theorem IsCompactOperator.image_subset_compact_of_isVonNBounded {f : M₁ →ₛₗ[σ₁₂] M₂}
    (hf : IsCompactOperator f) {S : Set M₁} (hS : IsVonNBounded 𝕜₁ S) :
    ∃ K : Set M₂, IsCompact K ∧ f '' S ⊆ K 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsVonNBounded 𝕜₁ S)
Goal: ∃ K, IsCompact K ∧ ⇑f '' S ⊆ K
-- IsCompactOperator.isCompact_closure_image_of_isVonNBounded
theorem IsCompactOperator.isCompact_closure_image_of_isVonNBounded [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂}
    (hf : IsCompactOperator f) {S : Set M₁} (hS : IsVonNBounded 𝕜₁ S) :
    IsCompact (closure <| f '' S) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsVonNBounded 𝕜₁ S)
Goal: IsCompact (closure (⇑f '' S))
-- IsCompactOperator.image_subset_compact_of_bounded
theorem IsCompactOperator.image_subset_compact_of_bounded [ContinuousConstSMul 𝕜₂ M₂]
    {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) {S : Set M₁} (hS : Bornology.IsBounded S) :
    ∃ K : Set M₂, IsCompact K ∧ f '' S ⊆ K 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsBounded S)
Goal: ∃ K, IsCompact K ∧ ⇑f '' S ⊆ K
-- IsCompactOperator.isCompact_closure_image_of_bounded
theorem IsCompactOperator.isCompact_closure_image_of_bounded [ContinuousConstSMul 𝕜₂ M₂]
    [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) {S : Set M₁}
    (hS : Bornology.IsBounded S) : IsCompact (closure <| f '' S) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) {S : Set M₁} (hS : Bornology.IsBounded S)
Goal: IsCompact (closure (⇑f '' S))
-- IsCompactOperator.image_ball_subset_compact
theorem IsCompactOperator.image_ball_subset_compact [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂}
    (hf : IsCompactOperator f) (r : ℝ) : ∃ K : Set M₂, IsCompact K ∧ f '' Metric.ball 0 r ⊆ K 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ)
Goal: ∃ K, IsCompact K ∧ ⇑f '' Metric.ball 0 r ⊆ K
-- IsCompactOperator.image_closedBall_subset_compact
theorem IsCompactOperator.image_closedBall_subset_compact [ContinuousConstSMul 𝕜₂ M₂]
    {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) (r : ℝ) :
    ∃ K : Set M₂, IsCompact K ∧ f '' Metric.closedBall 0 r ⊆ K 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ)
Goal: ∃ K, IsCompact K ∧ ⇑f '' Metric.closedBall 0 r ⊆ K
-- IsCompactOperator.isCompact_closure_image_ball
theorem IsCompactOperator.isCompact_closure_image_ball [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂]
    {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) (r : ℝ) :
    IsCompact (closure <| f '' Metric.ball 0 r) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ)
Goal: IsCompact (closure (⇑f '' Metric.ball 0 r))
-- IsCompactOperator.isCompact_closure_image_closedBall
theorem IsCompactOperator.isCompact_closure_image_closedBall [ContinuousConstSMul 𝕜₂ M₂]
    [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) (r : ℝ) :
    IsCompact (closure <| f '' Metric.closedBall 0 r) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f) (r : ℝ)
Goal: IsCompact (closure (⇑f '' Metric.closedBall 0 r))
-- isCompactOperator_iff_image_ball_subset_compact
theorem isCompactOperator_iff_image_ball_subset_compact [ContinuousConstSMul 𝕜₂ M₂]
    (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
    IsCompactOperator f ↔ ∃ K : Set M₂, IsCompact K ∧ f '' Metric.ball 0 r ⊆ K 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r)
Goal: IsCompactOperator ⇑f ↔ ∃ K, IsCompact K ∧ ⇑f '' Metric.ball 0 r ⊆ K
-- isCompactOperator_iff_image_closedBall_subset_compact
theorem isCompactOperator_iff_image_closedBall_subset_compact [ContinuousConstSMul 𝕜₂ M₂]
    (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
    IsCompactOperator f ↔ ∃ K : Set M₂, IsCompact K ∧ f '' Metric.closedBall 0 r ⊆ K 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r)
Goal: IsCompactOperator ⇑f ↔ ∃ K, IsCompact K ∧ ⇑f '' Metric.closedBall 0 r ⊆ K
-- isCompactOperator_iff_isCompact_closure_image_ball
theorem isCompactOperator_iff_isCompact_closure_image_ball [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂]
    (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
    IsCompactOperator f ↔ IsCompact (closure <| f '' Metric.ball 0 r) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r)
Goal: IsCompactOperator ⇑f ↔ IsCompact (closure (⇑f '' Metric.ball 0 r))
-- isCompactOperator_iff_isCompact_closure_image_closedBall
theorem isCompactOperator_iff_isCompact_closure_image_closedBall [ContinuousConstSMul 𝕜₂ M₂]
    [T2Space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
    IsCompactOperator f ↔ IsCompact (closure <| f '' Metric.closedBall 0 r) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [SeminormedRing 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r)
Goal: IsCompactOperator ⇑f ↔ IsCompact (closure (⇑f '' Metric.closedBall 0 r))
-- IsCompactOperator.smul
theorem IsCompactOperator.smul {S : Type*} [Monoid S] [DistribMulAction S M₂]
    [ContinuousConstSMul S M₂] {f : M₁ → M₂} (hf : IsCompactOperator f) (c : S) :
    IsCompactOperator (c • f) 
Arguments: {M₁ : Type u_5} {M₂ : Type u_6} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] {S : Type u_9} [Monoid S] [DistribMulAction S M₂] [ContinuousConstSMul S M₂] {f : M₁ → M₂} (hf : IsCompactOperator f) (c : S)
Goal: IsCompactOperator (c • f)
-- IsCompactOperator.add
theorem IsCompactOperator.add [ContinuousAdd M₂] {f g : M₁ → M₂} (hf : IsCompactOperator f)
    (hg : IsCompactOperator g) : IsCompactOperator (f + g) 
Arguments: {M₁ : Type u_5} {M₂ : Type u_6} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₂] [AddCommMonoid M₂] [ContinuousAdd M₂] {f g : M₁ → M₂} (hf : IsCompactOperator f) (hg : IsCompactOperator g)
Goal: IsCompactOperator (f + g)
-- IsCompactOperator.neg
theorem IsCompactOperator.neg [ContinuousNeg M₄] {f : M₁ → M₄} (hf : IsCompactOperator f) :
    IsCompactOperator (-f) 
Arguments: {M₁ : Type u_5} {M₄ : Type u_8} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₄] [AddCommGroup M₄] [ContinuousNeg M₄] {f : M₁ → M₄} (hf : IsCompactOperator f)
Goal: IsCompactOperator (-f)
-- IsCompactOperator.sub
theorem IsCompactOperator.sub [TopologicalAddGroup M₄] {f g : M₁ → M₄} (hf : IsCompactOperator f)
    (hg : IsCompactOperator g) : IsCompactOperator (f - g) 
Arguments: {M₁ : Type u_5} {M₄ : Type u_8} [TopologicalSpace M₁] [AddCommMonoid M₁] [TopologicalSpace M₄] [AddCommGroup M₄] [TopologicalAddGroup M₄] {f g : M₁ → M₄} (hf : IsCompactOperator f) (hg : IsCompactOperator g)
Goal: IsCompactOperator (f - g)
-- IsCompactOperator.comp_clm
theorem IsCompactOperator.comp_clm [AddCommMonoid M₂] [Module R₂ M₂] {f : M₂ → M₃}
    (hf : IsCompactOperator f) (g : M₁ →SL[σ₁₂] M₂) : IsCompactOperator (f ∘ g) 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [TopologicalSpace M₁] [TopologicalSpace M₂] [TopologicalSpace M₃] [AddCommMonoid M₁] [Module R₁ M₁] [AddCommMonoid M₂] [Module R₂ M₂] {f : M₂ → M₃} (hf : IsCompactOperator f) (g : M₁ →SL[σ₁₂] M₂)
Goal: IsCompactOperator (f ∘ ⇑g)
-- IsCompactOperator.continuous_comp
theorem IsCompactOperator.continuous_comp {f : M₁ → M₂} (hf : IsCompactOperator f) {g : M₂ → M₃}
    (hg : Continuous g) : IsCompactOperator (g ∘ f) 
Arguments: {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [TopologicalSpace M₁] [TopologicalSpace M₂] [TopologicalSpace M₃] [AddCommMonoid M₁] {f : M₁ → M₂} (hf : IsCompactOperator f) {g : M₂ → M₃} (hg : Continuous g)
Goal: IsCompactOperator (g ∘ f)
-- IsCompactOperator.clm_comp
theorem IsCompactOperator.clm_comp [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M₃]
    [Module R₃ M₃] {f : M₁ → M₂} (hf : IsCompactOperator f) (g : M₂ →SL[σ₂₃] M₃) :
    IsCompactOperator (g ∘ f) 
Arguments: {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₂] [Semiring R₃] {σ₂₃ : R₂ →+* R₃} {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [TopologicalSpace M₁] [TopologicalSpace M₂] [TopologicalSpace M₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M₃] [Module R₃ M₃] {f : M₁ → M₂} (hf : IsCompactOperator f) (g : M₂ →SL[σ₂₃] M₃)
Goal: IsCompactOperator (⇑g ∘ f)