ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- interior_subset_intrinsicInterior
theorem interior_subset_intrinsicInterior : interior s ⊆ intrinsicInterior 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: interior s ⊆ intrinsicInterior 𝕜 s
-- intrinsicClosure_subset_closure
theorem intrinsicClosure_subset_closure : intrinsicClosure 𝕜 s ⊆ closure s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: intrinsicClosure 𝕜 s ⊆ closure s
-- intrinsicFrontier_subset_frontier
theorem intrinsicFrontier_subset_frontier : intrinsicFrontier 𝕜 s ⊆ frontier s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: intrinsicFrontier 𝕜 s ⊆ frontier s
-- intrinsicClosure_subset_affineSpan
theorem intrinsicClosure_subset_affineSpan : intrinsicClosure 𝕜 s ⊆ affineSpan 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: intrinsicClosure 𝕜 s ⊆ ↑(affineSpan 𝕜 s)
-- intrinsicClosure_diff_intrinsicFrontier
@[simp]
theorem intrinsicClosure_diff_intrinsicFrontier (s : Set P) :
    intrinsicClosure 𝕜 s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (s : Set P)
Goal: intrinsicClosure 𝕜 s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s
-- intrinsicClosure_diff_intrinsicInterior
@[simp]
theorem intrinsicClosure_diff_intrinsicInterior (s : Set P) :
    intrinsicClosure 𝕜 s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (s : Set P)
Goal: intrinsicClosure 𝕜 s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s
-- intrinsicInterior_union_intrinsicFrontier
@[simp]
theorem intrinsicInterior_union_intrinsicFrontier (s : Set P) :
    intrinsicInterior 𝕜 s ∪ intrinsicFrontier 𝕜 s = intrinsicClosure 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (s : Set P)
Goal: intrinsicInterior 𝕜 s ∪ intrinsicFrontier 𝕜 s = intrinsicClosure 𝕜 s
-- intrinsicFrontier_union_intrinsicInterior
@[simp]
theorem intrinsicFrontier_union_intrinsicInterior (s : Set P) :
    intrinsicFrontier 𝕜 s ∪ intrinsicInterior 𝕜 s = intrinsicClosure 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (s : Set P)
Goal: intrinsicFrontier 𝕜 s ∪ intrinsicInterior 𝕜 s = intrinsicClosure 𝕜 s
-- isClosed_intrinsicClosure
theorem isClosed_intrinsicClosure (hs : IsClosed (affineSpan 𝕜 s : Set P)) :
    IsClosed (intrinsicClosure 𝕜 s) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} (hs : IsClosed ↑(affineSpan 𝕜 s))
Goal: IsClosed (intrinsicClosure 𝕜 s)
-- isClosed_intrinsicFrontier
theorem isClosed_intrinsicFrontier (hs : IsClosed (affineSpan 𝕜 s : Set P)) :
    IsClosed (intrinsicFrontier 𝕜 s) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} (hs : IsClosed ↑(affineSpan 𝕜 s))
Goal: IsClosed (intrinsicFrontier 𝕜 s)
-- affineSpan_intrinsicClosure
@[simp]
theorem affineSpan_intrinsicClosure (s : Set P) :
    affineSpan 𝕜 (intrinsicClosure 𝕜 s) = affineSpan 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (s : Set P)
Goal: affineSpan 𝕜 (intrinsicClosure 𝕜 s) = affineSpan 𝕜 s
-- IsClosed.intrinsicClosure
protected theorem IsClosed.intrinsicClosure (hs : IsClosed ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s)) :
    intrinsicClosure 𝕜 s = s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} (hs : IsClosed (Subtype.val ⁻¹' s))
Goal: intrinsicClosure 𝕜 s = s
-- intrinsicClosure_idem
@[simp]
theorem intrinsicClosure_idem (s : Set P) :
    intrinsicClosure 𝕜 (intrinsicClosure 𝕜 s) = intrinsicClosure 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (s : Set P)
Goal: intrinsicClosure 𝕜 (intrinsicClosure 𝕜 s) = intrinsicClosure 𝕜 s
-- AffineIsometry.image_intrinsicInterior
@[simp]
theorem image_intrinsicInterior (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
    intrinsicInterior 𝕜 (φ '' s) = φ '' intrinsicInterior 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {W : Type u_3} {Q : Type u_4} {P : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [MetricSpace P] [PseudoMetricSpace Q] [NormedAddTorsor V P] [NormedAddTorsor W Q] (φ : P →ᵃⁱ[𝕜] Q) (s : Set P)
Goal: intrinsicInterior 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicInterior 𝕜 s
-- AffineIsometry.image_intrinsicFrontier
@[simp]
theorem image_intrinsicFrontier (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
    intrinsicFrontier 𝕜 (φ '' s) = φ '' intrinsicFrontier 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {W : Type u_3} {Q : Type u_4} {P : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [MetricSpace P] [PseudoMetricSpace Q] [NormedAddTorsor V P] [NormedAddTorsor W Q] (φ : P →ᵃⁱ[𝕜] Q) (s : Set P)
Goal: intrinsicFrontier 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicFrontier 𝕜 s
-- AffineIsometry.image_intrinsicClosure
@[simp]
theorem image_intrinsicClosure (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
    intrinsicClosure 𝕜 (φ '' s) = φ '' intrinsicClosure 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {W : Type u_3} {Q : Type u_4} {P : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [MetricSpace P] [PseudoMetricSpace Q] [NormedAddTorsor V P] [NormedAddTorsor W Q] (φ : P →ᵃⁱ[𝕜] Q) (s : Set P)
Goal: intrinsicClosure 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicClosure 𝕜 s
-- intrinsicClosure_eq_closure
@[simp]
theorem intrinsicClosure_eq_closure : intrinsicClosure 𝕜 s = closure s 
Arguments: (𝕜 : Type u_1) {V : Type u_2} {P : Type u_5} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [FiniteDimensional 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : Set P)
Goal: intrinsicClosure 𝕜 s = closure s
-- closure_diff_intrinsicInterior
@[simp]
theorem closure_diff_intrinsicInterior (s : Set P) :
    closure s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [FiniteDimensional 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : Set P)
Goal: closure s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s
-- closure_diff_intrinsicFrontier
@[simp]
theorem closure_diff_intrinsicFrontier (s : Set P) :
    closure s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [FiniteDimensional 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : Set P)
Goal: closure s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s
-- _private.Mathlib.Analysis.Convex.Intrinsic.0.aux
private theorem aux {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] (φ : α ≃ₜ β)
    (s : Set β) : (interior s).Nonempty ↔ (interior (φ ⁻¹' s)).Nonempty 
Arguments: {α : Type u_6} {β : Type u_7} [TopologicalSpace α] [TopologicalSpace β] (φ : α ≃ₜ β) (s : Set β)
Goal: (interior s).Nonempty ↔ (interior (⇑φ ⁻¹' s)).Nonempty
-- Set.Nonempty.intrinsicInterior
protected theorem Set.Nonempty.intrinsicInterior (hscv : Convex ℝ s) (hsne : s.Nonempty) :
    (intrinsicInterior ℝ s).Nonempty 
Arguments: {V : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {s : Set V} (hscv : Convex ℝ s) (hsne : s.Nonempty)
Goal: (intrinsicInterior ℝ s).Nonempty
-- intrinsicInterior_nonempty
theorem intrinsicInterior_nonempty (hs : Convex ℝ s) :
    (intrinsicInterior ℝ s).Nonempty ↔ s.Nonempty 
Arguments: {V : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {s : Set V} (hs : Convex ℝ s)
Goal: (intrinsicInterior ℝ s).Nonempty ↔ s.Nonempty
-- DoubleCentralizer.central
structure DoubleCentralizer (𝕜 : Type u) (A : Type v) [NontriviallyNormedField 𝕜]
    [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] extends
    (A →L[𝕜] A) × (A →L[𝕜] A) where
    central : ∀ x y : A, snd x * y = x * fst y
Arguments: {𝕜 : Type u} {A : Type v} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (self : 𝓜(𝕜, A)) (x y : A)
Goal: self.toProd.2 x * y = x * self.toProd.1 y
-- DoubleCentralizer.ext
@[ext]
lemma DoubleCentralizer.ext (𝕜 : Type u) (A : Type v) [NontriviallyNormedField 𝕜]
    [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
    (a b : 𝓜(𝕜, A)) (h : a.toProd = b.toProd) : a = b 
Arguments: (𝕜 : Type u) (A : Type v) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A)) (h : a.toProd = b.toProd)
Goal: a = b
-- DoubleCentralizer.range_toProd
theorem range_toProd :
    Set.range toProd = { lr : (A →L[𝕜] A) × (A →L[𝕜] A) | ∀ x y, lr.2 x * y = x * lr.1 y } 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: Set.range DoubleCentralizer.toProd = {lr | ∀ (x y : A), lr.2 x * y = x * lr.1 y}
-- DoubleCentralizer.smul_toProd
@[simp]
theorem smul_toProd (s : S) (a : 𝓜(𝕜, A)) : (s • a).toProd = s • a.toProd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] {S : Type u_3} [Monoid S] [DistribMulAction S A] [SMulCommClass 𝕜 S A] [ContinuousConstSMul S A] [IsScalarTower S A A] [SMulCommClass S A A] (s : S) (a : 𝓜(𝕜, A))
Goal: (s • a).toProd = s • a.toProd
-- DoubleCentralizer.smul_fst
theorem smul_fst (s : S) (a : 𝓜(𝕜, A)) : (s • a).fst = s • a.fst 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] {S : Type u_3} [Monoid S] [DistribMulAction S A] [SMulCommClass 𝕜 S A] [ContinuousConstSMul S A] [IsScalarTower S A A] [SMulCommClass S A A] (s : S) (a : 𝓜(𝕜, A))
Goal: (s • a).toProd.1 = s • a.toProd.1
-- DoubleCentralizer.smul_snd
theorem smul_snd (s : S) (a : 𝓜(𝕜, A)) : (s • a).snd = s • a.snd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] {S : Type u_3} [Monoid S] [DistribMulAction S A] [SMulCommClass 𝕜 S A] [ContinuousConstSMul S A] [IsScalarTower S A A] [SMulCommClass S A A] (s : S) (a : 𝓜(𝕜, A))
Goal: (s • a).toProd.2 = s • a.toProd.2
-- DoubleCentralizer.add_toProd
@[simp]
theorem add_toProd (a b : 𝓜(𝕜, A)) : (a + b).toProd = a.toProd + b.toProd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a + b).toProd = a.toProd + b.toProd
-- DoubleCentralizer.zero_toProd
@[simp]
theorem zero_toProd : (0 : 𝓜(𝕜, A)).toProd = 0 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: DoubleCentralizer.toProd 0 = 0
-- DoubleCentralizer.neg_toProd
@[simp]
theorem neg_toProd (a : 𝓜(𝕜, A)) : (-a).toProd = -a.toProd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: (-a).toProd = -a.toProd
-- DoubleCentralizer.sub_toProd
@[simp]
theorem sub_toProd (a b : 𝓜(𝕜, A)) : (a - b).toProd = a.toProd - b.toProd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a - b).toProd = a.toProd - b.toProd
-- DoubleCentralizer.one_toProd
@[simp]
theorem one_toProd : (1 : 𝓜(𝕜, A)).toProd = 1 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: DoubleCentralizer.toProd 1 = 1
-- DoubleCentralizer.natCast_toProd
@[simp]
theorem natCast_toProd (n : ℕ) : (n : 𝓜(𝕜, A)).toProd = n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℕ)
Goal: (↑n).toProd = ↑n
-- DoubleCentralizer.intCast_toProd
@[simp]
theorem intCast_toProd (n : ℤ) : (n : 𝓜(𝕜, A)).toProd = n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℤ)
Goal: (↑n).toProd = ↑n
-- DoubleCentralizer.pow_toProd
@[simp]
theorem pow_toProd (n : ℕ) (a : 𝓜(𝕜, A)) : (a ^ n).toProd = a.toProd ^ n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℕ) (a : 𝓜(𝕜, A))
Goal: (a ^ n).toProd = a.toProd ^ n
-- DoubleCentralizer.add_fst
theorem add_fst (a b : 𝓜(𝕜, A)) : (a + b).fst = a.fst + b.fst 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a + b).toProd.1 = a.toProd.1 + b.toProd.1
-- DoubleCentralizer.add_snd
theorem add_snd (a b : 𝓜(𝕜, A)) : (a + b).snd = a.snd + b.snd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a + b).toProd.2 = a.toProd.2 + b.toProd.2
-- DoubleCentralizer.zero_fst
theorem zero_fst : (0 : 𝓜(𝕜, A)).fst = 0 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: (DoubleCentralizer.toProd 0).1 = 0
-- DoubleCentralizer.zero_snd
theorem zero_snd : (0 : 𝓜(𝕜, A)).snd = 0 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: (DoubleCentralizer.toProd 0).2 = 0
-- DoubleCentralizer.neg_fst
theorem neg_fst (a : 𝓜(𝕜, A)) : (-a).fst = -a.fst 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: (-a).toProd.1 = -a.toProd.1
-- DoubleCentralizer.neg_snd
theorem neg_snd (a : 𝓜(𝕜, A)) : (-a).snd = -a.snd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: (-a).toProd.2 = -a.toProd.2
-- DoubleCentralizer.sub_fst
theorem sub_fst (a b : 𝓜(𝕜, A)) : (a - b).fst = a.fst - b.fst 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a - b).toProd.1 = a.toProd.1 - b.toProd.1
-- DoubleCentralizer.sub_snd
theorem sub_snd (a b : 𝓜(𝕜, A)) : (a - b).snd = a.snd - b.snd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a - b).toProd.2 = a.toProd.2 - b.toProd.2
-- DoubleCentralizer.one_fst
theorem one_fst : (1 : 𝓜(𝕜, A)).fst = 1 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: (DoubleCentralizer.toProd 1).1 = 1
-- DoubleCentralizer.one_snd
theorem one_snd : (1 : 𝓜(𝕜, A)).snd = 1 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: (DoubleCentralizer.toProd 1).2 = 1
-- DoubleCentralizer.mul_fst
@[simp]
theorem mul_fst (a b : 𝓜(𝕜, A)) : (a * b).fst = a.fst * b.fst 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a * b).toProd.1 = a.toProd.1 * b.toProd.1
-- DoubleCentralizer.mul_snd
@[simp]
theorem mul_snd (a b : 𝓜(𝕜, A)) : (a * b).snd = b.snd * a.snd 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : 𝓜(𝕜, A))
Goal: (a * b).toProd.2 = b.toProd.2 * a.toProd.2
-- DoubleCentralizer.natCast_fst
theorem natCast_fst (n : ℕ) : (n : 𝓜(𝕜, A)).fst = n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℕ)
Goal: (↑n).toProd.1 = ↑n
-- DoubleCentralizer.natCast_snd
theorem natCast_snd (n : ℕ) : (n : 𝓜(𝕜, A)).snd = n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℕ)
Goal: (↑n).toProd.2 = ↑n
-- DoubleCentralizer.intCast_fst
theorem intCast_fst (n : ℤ) : (n : 𝓜(𝕜, A)).fst = n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℤ)
Goal: (↑n).toProd.1 = ↑n
-- DoubleCentralizer.intCast_snd
theorem intCast_snd (n : ℤ) : (n : 𝓜(𝕜, A)).snd = n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℤ)
Goal: (↑n).toProd.2 = ↑n
-- DoubleCentralizer.pow_fst
theorem pow_fst (n : ℕ) (a : 𝓜(𝕜, A)) : (a ^ n).fst = a.fst ^ n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℕ) (a : 𝓜(𝕜, A))
Goal: (a ^ n).toProd.1 = a.toProd.1 ^ n
-- DoubleCentralizer.pow_snd
theorem pow_snd (n : ℕ) (a : 𝓜(𝕜, A)) : (a ^ n).snd = a.snd ^ n 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (n : ℕ) (a : 𝓜(𝕜, A))
Goal: (a ^ n).toProd.2 = a.toProd.2 ^ n
-- DoubleCentralizer.toProdMulOpposite_injective
theorem toProdMulOpposite_injective :
    Function.Injective (toProdMulOpposite : 𝓜(𝕜, A) → (A →L[𝕜] A) × (A →L[𝕜] A)ᵐᵒᵖ) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: Function.Injective DoubleCentralizer.toProdMulOpposite
-- DoubleCentralizer.range_toProdMulOpposite
theorem range_toProdMulOpposite :
    Set.range toProdMulOpposite =
      { lr : (A →L[𝕜] A) × (A →L[𝕜] A)ᵐᵒᵖ | ∀ x y, unop lr.2 x * y = x * lr.1 y } 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: Set.range DoubleCentralizer.toProdMulOpposite = {lr | ∀ (x y : A), (MulOpposite.unop lr.2) x * y = x * lr.1 y}
-- DoubleCentralizer.toProdHom_apply
@[simps]
def toProdHom : 𝓜(𝕜, A) →+ (A →L[𝕜] A) × (A →L[𝕜] A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (self : 𝓜(𝕜, A))
Goal: DoubleCentralizer.toProdHom self = self.toProd
-- DoubleCentralizer.toProdMulOppositeHom_apply
@[simps]
def toProdMulOppositeHom : 𝓜(𝕜, A) →+* (A →L[𝕜] A) × (A →L[𝕜] A)ᵐᵒᵖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: ∀ (a : 𝓜(𝕜, A)), DoubleCentralizer.toProdMulOppositeHom a = a.toProdMulOpposite
-- DoubleCentralizer.algebraMap_toProd
@[simp]
theorem algebraMap_toProd (k : 𝕜) : (algebraMap 𝕜 𝓜(𝕜, A) k).toProd = algebraMap 𝕜 _ k 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (k : 𝕜)
Goal: ((algebraMap 𝕜 𝓜(𝕜, A)) k).toProd = (algebraMap 𝕜 ((A →L[𝕜] A) × (A →L[𝕜] A))) k
-- DoubleCentralizer.algebraMap_fst
theorem algebraMap_fst (k : 𝕜) : (algebraMap 𝕜 𝓜(𝕜, A) k).fst = algebraMap 𝕜 _ k 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (k : 𝕜)
Goal: ((algebraMap 𝕜 𝓜(𝕜, A)) k).toProd.1 = (algebraMap 𝕜 (A →L[𝕜] A)) k
-- DoubleCentralizer.algebraMap_snd
theorem algebraMap_snd (k : 𝕜) : (algebraMap 𝕜 𝓜(𝕜, A) k).snd = algebraMap 𝕜 _ k 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (k : 𝕜)
Goal: ((algebraMap 𝕜 𝓜(𝕜, A)) k).toProd.2 = (algebraMap 𝕜 (A →L[𝕜] A)) k
-- DoubleCentralizer.star_fst
@[simp]
theorem star_fst (a : 𝓜(𝕜, A)) (b : A) : (star a).fst b = star (a.snd (star b)) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing 𝕜] [StarRing A] [StarModule 𝕜 A] [NormedStarGroup A] (a : 𝓜(𝕜, A)) (b : A)
Goal: (star a).toProd.1 b = star (a.toProd.2 (star b))
-- DoubleCentralizer.star_snd
@[simp]
theorem star_snd (a : 𝓜(𝕜, A)) (b : A) : (star a).snd b = star (a.fst (star b)) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing 𝕜] [StarRing A] [StarModule 𝕜 A] [NormedStarGroup A] (a : 𝓜(𝕜, A)) (b : A)
Goal: (star a).toProd.2 b = star (a.toProd.1 (star b))
-- DoubleCentralizer.coe_fst
@[simp, norm_cast]
theorem coe_fst (a : A) : (a : 𝓜(𝕜, A)).fst = ContinuousLinearMap.mul 𝕜 A a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : A)
Goal: (↑𝕜 a).toProd.1 = (ContinuousLinearMap.mul 𝕜 A) a
-- DoubleCentralizer.coe_snd
@[simp, norm_cast]
theorem coe_snd (a : A) : (a : 𝓜(𝕜, A)).snd = (ContinuousLinearMap.mul 𝕜 A).flip a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : A)
Goal: (↑𝕜 a).toProd.2 = (ContinuousLinearMap.mul 𝕜 A).flip a
-- DoubleCentralizer.coe_eq_algebraMap
theorem coe_eq_algebraMap : (DoubleCentralizer.coe 𝕜 : 𝕜 → 𝓜(𝕜, 𝕜)) = algebraMap 𝕜 𝓜(𝕜, 𝕜) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜]
Goal: ↑𝕜 = ⇑(algebraMap 𝕜 𝓜(𝕜, 𝕜))
-- DoubleCentralizer.coeHom_apply
@[simps]
noncomputable def coeHom [StarRing 𝕜] [StarRing A] [StarModule 𝕜 A] [NormedStarGroup A] :
    A →⋆ₙₐ[𝕜] 𝓜(𝕜, A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing 𝕜] [StarRing A] [StarModule 𝕜 A] [NormedStarGroup A] (a : A)
Goal: DoubleCentralizer.coeHom a = ↑𝕜 a
-- DoubleCentralizer.norm_def
theorem norm_def (a : 𝓜(𝕜, A)) : ‖a‖ = ‖toProdHom a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: ‖a‖ = ‖DoubleCentralizer.toProdHom a‖
-- DoubleCentralizer.nnnorm_def
theorem nnnorm_def (a : 𝓜(𝕜, A)) : ‖a‖₊ = ‖toProdHom a‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: ‖a‖₊ = ‖DoubleCentralizer.toProdHom a‖₊