ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoundedContinuousFunction.coe_neg
@[simp]
theorem coe_neg : ⇑(-f) = -f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ⇑(-f) = -⇑f
-- BoundedContinuousFunction.neg_apply
theorem neg_apply : (-f) x = -f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) {x : α}
Goal: (-f) x = -f x
-- BoundedContinuousFunction.mkOfCompact_neg
@[simp]
theorem mkOfCompact_neg [CompactSpace α] (f : C(α, β)) : mkOfCompact (-f) = -mkOfCompact f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [CompactSpace α] (f : C(α, β))
Goal: BoundedContinuousFunction.mkOfCompact (-f) = -BoundedContinuousFunction.mkOfCompact f
-- BoundedContinuousFunction.mkOfCompact_sub
@[simp]
theorem mkOfCompact_sub [CompactSpace α] (f g : C(α, β)) :
    mkOfCompact (f - g) = mkOfCompact f - mkOfCompact g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [CompactSpace α] (f g : C(α, β))
Goal: BoundedContinuousFunction.mkOfCompact (f - g) =
  BoundedContinuousFunction.mkOfCompact f - BoundedContinuousFunction.mkOfCompact g
-- BoundedContinuousFunction.coe_zsmulRec
@[simp]
theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec (· • ·) z f) = z • ⇑f
  
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (z : ℤ)
Goal: ⇑(zsmulRec (fun x x_1 => x • x_1) z f) = z • ⇑f
-- BoundedContinuousFunction.coe_zsmul
@[simp]
theorem coe_zsmul (r : ℤ) (f : α →ᵇ β) : ⇑(r • f) = r • ⇑f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (r : ℤ) (f : BoundedContinuousFunction α β)
Goal: ⇑(r • f) = r • ⇑f
-- BoundedContinuousFunction.zsmul_apply
@[simp]
theorem zsmul_apply (r : ℤ) (f : α →ᵇ β) (v : α) : (r • f) v = r • f v 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (r : ℤ) (f : BoundedContinuousFunction α β) (v : α)
Goal: (r • f) v = r • f v
-- BoundedContinuousFunction.nnnorm_def
theorem nnnorm_def : ‖f‖₊ = nndist f 0 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ‖f‖₊ = nndist f 0
-- BoundedContinuousFunction.nnnorm_coe_le_nnnorm
theorem nnnorm_coe_le_nnnorm (x : α) : ‖f x‖₊ ≤ ‖f‖₊ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (x : α)
Goal: ‖f x‖₊ ≤ ‖f‖₊
-- BoundedContinuousFunction.nndist_le_two_nnnorm
theorem nndist_le_two_nnnorm (x y : α) : nndist (f x) (f y) ≤ 2 * ‖f‖₊ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (x y : α)
Goal: nndist (f x) (f y) ≤ 2 * ‖f‖₊
-- BoundedContinuousFunction.nnnorm_le
theorem nnnorm_le (C : ℝ≥0) : ‖f‖₊ ≤ C ↔ ∀ x : α, ‖f x‖₊ ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (C : ℝ≥0)
Goal: ‖f‖₊ ≤ C ↔ ∀ (x : α), ‖f x‖₊ ≤ C
-- BoundedContinuousFunction.nnnorm_const_le
theorem nnnorm_const_le (b : β) : ‖const α b‖₊ ≤ ‖b‖₊ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (b : β)
Goal: ‖BoundedContinuousFunction.const α b‖₊ ≤ ‖b‖₊
-- BoundedContinuousFunction.nnnorm_const_eq
@[simp]
theorem nnnorm_const_eq [Nonempty α] (b : β) : ‖const α b‖₊ = ‖b‖₊ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [Nonempty α] (b : β)
Goal: ‖BoundedContinuousFunction.const α b‖₊ = ‖b‖₊
-- BoundedContinuousFunction.nnnorm_eq_iSup_nnnorm
theorem nnnorm_eq_iSup_nnnorm : ‖f‖₊ = ⨆ x : α, ‖f x‖₊ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ‖f‖₊ = ⨆ x, ‖f x‖₊
-- BoundedContinuousFunction.abs_diff_coe_le_dist
theorem abs_diff_coe_le_dist : ‖f x - g x‖ ≤ dist f g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f g : BoundedContinuousFunction α β) {x : α}
Goal: ‖f x - g x‖ ≤ dist f g
-- BoundedContinuousFunction.coe_le_coe_add_dist
theorem coe_le_coe_add_dist {f g : α →ᵇ ℝ} : f x ≤ g x + dist f g 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {f g : BoundedContinuousFunction α ℝ}
Goal: f x ≤ g x + dist f g
-- BoundedContinuousFunction.norm_compContinuous_le
theorem norm_compContinuous_le [TopologicalSpace γ] (f : α →ᵇ β) (g : C(γ, α)) :
    ‖f.compContinuous g‖ ≤ ‖f‖ 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [SeminormedAddCommGroup β] [TopologicalSpace γ] (f : BoundedContinuousFunction α β) (g : C(γ, α))
Goal: ‖f.compContinuous g‖ ≤ ‖f‖
-- BoundedContinuousFunction.coe_smul
@[simp]
theorem coe_smul (c : 𝕜) (f : α →ᵇ β) : ⇑(c • f) = fun x => c • f x 
Arguments: {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) (f : BoundedContinuousFunction α β)
Goal: ⇑(c • f) = fun x => c • f x
-- BoundedContinuousFunction.smul_apply
theorem smul_apply (c : 𝕜) (f : α →ᵇ β) (x : α) : (c • f) x = c • f x 
Arguments: {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) (f : BoundedContinuousFunction α β) (x : α)
Goal: (c • f) x = c • f x
-- BoundedContinuousFunction.evalCLM_apply
@[simp]
theorem evalCLM_apply (x : α) (f : α →ᵇ β) : evalCLM 𝕜 x f = f x 
Arguments: {α : Type u} {β : Type v} (𝕜 : Type u_2) [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Semiring 𝕜] [AddCommMonoid β] [Module 𝕜 β] [BoundedSMul 𝕜 β] [BoundedAdd β] [ContinuousAdd β] (x : α) (f : BoundedContinuousFunction α β)
Goal: (BoundedContinuousFunction.evalCLM 𝕜 x) f = f x
-- BoundedContinuousFunction.toContinuousMapLinearMap_apply
@[simps]
def toContinuousMapLinearMap : (α →ᵇ β) →ₗ[𝕜] C(α, β) 
Arguments: (α : Type u) (β : Type v) (𝕜 : Type u_2) [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Semiring 𝕜] [AddCommMonoid β] [Module 𝕜 β] [BoundedSMul 𝕜 β] [BoundedAdd β] [ContinuousAdd β] (self : BoundedContinuousFunction α β)
Goal: (BoundedContinuousFunction.toContinuousMapLinearMap α β 𝕜) self = self.toContinuousMap
-- ContinuousLinearMap.compLeftContinuousBounded_apply
@[simp]
theorem _root_.ContinuousLinearMap.compLeftContinuousBounded_apply (g : β →L[𝕜] γ) (f : α →ᵇ β)
    (x : α) : (g.compLeftContinuousBounded α f) x = g (f x) 
Arguments: (α : Type u) {β : Type v} {γ : Type w} {𝕜 : Type u_2} [TopologicalSpace α] [SeminormedAddCommGroup β] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 β] [SeminormedAddCommGroup γ] [NormedSpace 𝕜 γ] (g : β →L[𝕜] γ) (f : BoundedContinuousFunction α β) (x : α)
Goal: ((ContinuousLinearMap.compLeftContinuousBounded α g) f) x = g (f x)
-- BoundedContinuousFunction.coe_npowRec
@[simp]
theorem coe_npowRec (f : α →ᵇ R) : ∀ n, ⇑(npowRec n f) = (⇑f) ^ n
  
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (f : BoundedContinuousFunction α R) (n : ℕ)
Goal: ⇑(npowRec n f) = ⇑f ^ n
-- BoundedContinuousFunction.coe_natCast
@[simp, norm_cast]
theorem coe_natCast (n : ℕ) : ((n : α →ᵇ R) : α → R) = n 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ℕ)
Goal: ⇑↑n = ↑n
-- BoundedContinuousFunction.coe_ofNat
@[simp, norm_cast]
theorem coe_ofNat (n : ℕ) [n.AtLeastTwo] :
    ((no_index OfNat.ofNat n : α →ᵇ R) : α → R) = OfNat.ofNat n 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ℕ) [n.AtLeastTwo]
Goal: ⇑(OfNat.ofNat n) = OfNat.ofNat n
-- BoundedContinuousFunction.coe_intCast
@[simp, norm_cast]
theorem coe_intCast (n : ℤ) : ((n : α →ᵇ R) : α → R) = n 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ℤ)
Goal: ⇑↑n = ↑n
-- BoundedContinuousFunction.algebraMap_apply
@[simp]
theorem algebraMap_apply (k : 𝕜) (a : α) : algebraMap 𝕜 (α →ᵇ γ) k a = k • (1 : γ) 
Arguments: {α : Type u} {γ : Type w} {𝕜 : Type u_2} [NormedField 𝕜] [TopologicalSpace α] [NormedRing γ] [NormedAlgebra 𝕜 γ] (k : 𝕜) (a : α)
Goal: ((algebraMap 𝕜 (BoundedContinuousFunction α γ)) k) a = k • 1
-- BoundedContinuousFunction.NNReal.upper_bound
theorem NNReal.upper_bound {α : Type*} [TopologicalSpace α] (f : α →ᵇ ℝ≥0) (x : α) :
    f x ≤ nndist f 0 
Arguments: {α : Type u_2} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ≥0) (x : α)
Goal: f x ≤ nndist f 0
-- BoundedContinuousFunction.coe_star
@[simp]
theorem coe_star (f : α →ᵇ β) : ⇑(star f) = star (⇑f) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β] (f : BoundedContinuousFunction α β)
Goal: ⇑(star f) = star ⇑f
-- BoundedContinuousFunction.star_apply
@[simp]
theorem star_apply (f : α →ᵇ β) (x : α) : star f x = star (f x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β] (f : BoundedContinuousFunction α β) (x : α)
Goal: (star f) x = star (f x)
-- BoundedContinuousFunction.coe_sup
@[simp, norm_cast] lemma coe_sup (f g : α →ᵇ β) : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f g : BoundedContinuousFunction α β)
Goal: ⇑(f ⊔ g) = ⇑f ⊔ ⇑g
-- BoundedContinuousFunction.coe_inf
@[simp, norm_cast] lemma coe_inf (f g : α →ᵇ β) : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f g : BoundedContinuousFunction α β)
Goal: ⇑(f ⊓ g) = ⇑f ⊓ ⇑g
-- BoundedContinuousFunction.coe_abs
@[simp, norm_cast] lemma coe_abs (f : α →ᵇ β) : ⇑|f| = |⇑f| 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ⇑|f| = |⇑f|
-- BoundedContinuousFunction.coe_posPart
@[simp, norm_cast] lemma coe_posPart (f : α →ᵇ β) : ⇑f⁺ = (⇑f)⁺ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ⇑f⁺ = (⇑f)⁺
-- BoundedContinuousFunction.coe_negPart
@[simp, norm_cast] lemma coe_negPart (f : α →ᵇ β) : ⇑f⁻ = (⇑f)⁻ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ⇑f⁻ = (⇑f)⁻
-- BoundedContinuousFunction.nnrealPart_coeFn_eq
@[simp]
theorem nnrealPart_coeFn_eq (f : α →ᵇ ℝ) : ⇑f.nnrealPart = Real.toNNReal ∘ ⇑f 
Arguments: {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ)
Goal: ⇑f.nnrealPart = Real.toNNReal ∘ ⇑f
-- BoundedContinuousFunction.nnnorm_coeFn_eq
@[simp]
theorem nnnorm_coeFn_eq (f : α →ᵇ ℝ) : ⇑f.nnnorm = NNNorm.nnnorm ∘ ⇑f 
Arguments: {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ)
Goal: ⇑f.nnnorm = nnnorm ∘ ⇑f
-- BoundedContinuousFunction.self_eq_nnrealPart_sub_nnrealPart_neg
theorem self_eq_nnrealPart_sub_nnrealPart_neg (f : α →ᵇ ℝ) :
    ⇑f = (↑) ∘ f.nnrealPart - (↑) ∘ (-f).nnrealPart 
Arguments: {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ)
Goal: ⇑f = NNReal.toReal ∘ ⇑f.nnrealPart - NNReal.toReal ∘ ⇑(-f).nnrealPart
-- BoundedContinuousFunction.abs_self_eq_nnrealPart_add_nnrealPart_neg
theorem abs_self_eq_nnrealPart_add_nnrealPart_neg (f : α →ᵇ ℝ) :
    abs ∘ ⇑f = (↑) ∘ f.nnrealPart + (↑) ∘ (-f).nnrealPart 
Arguments: {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ)
Goal: abs ∘ ⇑f = NNReal.toReal ∘ ⇑f.nnrealPart + NNReal.toReal ∘ ⇑(-f).nnrealPart
-- BoundedContinuousFunction.add_norm_nonneg
lemma add_norm_nonneg (f : α →ᵇ ℝ) :
    0 ≤ f + const _ ‖f‖ 
Arguments: {α : Type u_2} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ)
Goal: 0 ≤ f + BoundedContinuousFunction.const α ‖f‖
-- BoundedContinuousFunction.norm_sub_nonneg
lemma norm_sub_nonneg (f : α →ᵇ ℝ) :
    0 ≤ const _ ‖f‖ - f 
Arguments: {α : Type u_2} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ)
Goal: 0 ≤ BoundedContinuousFunction.const α ‖f‖ - f
-- ContinuousLinearEquiv.det_coe_symm
@[simp]
theorem det_coe_symm {R : Type*} [Field R] {M : Type*} [TopologicalSpace M] [AddCommGroup M]
    [Module R M] (A : M ≃L[R] M) : (A.symm : M →L[R] M).det = (A : M →L[R] M).det⁻¹ 
Arguments: {R : Type u_1} [Field R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] (A : M ≃L[R] M)
Goal: (↑A.symm).det = (↑A).det⁻¹
-- UniformFun.toFun_zero
@[to_additive (attr := simp)]
lemma UniformFun.toFun_one [One β] : toFun (1 : α →ᵤ β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β]
Goal: UniformFun.toFun 0 = 0
-- UniformFun.toFun_one
@[to_additive (attr := simp)]
lemma UniformFun.toFun_one [One β] : toFun (1 : α →ᵤ β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [One β]
Goal: UniformFun.toFun 1 = 1
-- UniformFun.ofFun_one
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_one [One β] : ofFun (1 : α → β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [One β]
Goal: UniformFun.ofFun 1 = 1
-- UniformFun.ofFun_zero
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_one [One β] : ofFun (1 : α → β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β]
Goal: UniformFun.ofFun 0 = 0
-- UniformOnFun.toFun_one
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_one [One β] : toFun 𝔖 (1 : α →ᵤ[𝔖] β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [One β]
Goal: (UniformOnFun.toFun 𝔖) 1 = 1
-- UniformOnFun.toFun_zero
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_one [One β] : toFun 𝔖 (1 : α →ᵤ[𝔖] β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Zero β]
Goal: (UniformOnFun.toFun 𝔖) 0 = 0
-- UniformOnFun.zero_apply
@[to_additive (attr := simp)]
lemma UniformOnFun.one_apply [One β] : ofFun 𝔖 (1 : α → β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Zero β]
Goal: (UniformOnFun.ofFun 𝔖) 0 = 0
-- UniformOnFun.one_apply
@[to_additive (attr := simp)]
lemma UniformOnFun.one_apply [One β] : ofFun 𝔖 (1 : α → β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [One β]
Goal: (UniformOnFun.ofFun 𝔖) 1 = 1
-- UniformFun.toFun_mul
@[to_additive (attr := simp)]
lemma UniformFun.toFun_mul [Mul β] (f g : α →ᵤ β) : toFun (f * g) = toFun f * toFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Mul β] (f g : α →ᵤ β)
Goal: UniformFun.toFun (f * g) = UniformFun.toFun f * UniformFun.toFun g
-- UniformFun.toFun_add
@[to_additive (attr := simp)]
lemma UniformFun.toFun_mul [Mul β] (f g : α →ᵤ β) : toFun (f * g) = toFun f * toFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Add β] (f g : α →ᵤ β)
Goal: UniformFun.toFun (f + g) = UniformFun.toFun f + UniformFun.toFun g
-- UniformFun.ofFun_mul
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_mul [Mul β] (f g : α → β) : ofFun (f * g) = ofFun f * ofFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Mul β] (f g : α → β)
Goal: UniformFun.ofFun (f * g) = UniformFun.ofFun f * UniformFun.ofFun g
-- UniformFun.ofFun_add
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_mul [Mul β] (f g : α → β) : ofFun (f * g) = ofFun f * ofFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Add β] (f g : α → β)
Goal: UniformFun.ofFun (f + g) = UniformFun.ofFun f + UniformFun.ofFun g
-- UniformOnFun.toFun_add
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_mul [Mul β] (f g : α →ᵤ[𝔖] β) :
    toFun 𝔖 (f * g) = toFun 𝔖 f * toFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Add β] (f g : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) (f + g) = (UniformOnFun.toFun 𝔖) f + (UniformOnFun.toFun 𝔖) g
-- UniformOnFun.toFun_mul
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_mul [Mul β] (f g : α →ᵤ[𝔖] β) :
    toFun 𝔖 (f * g) = toFun 𝔖 f * toFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Mul β] (f g : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) (f * g) = (UniformOnFun.toFun 𝔖) f * (UniformOnFun.toFun 𝔖) g
-- UniformOnFun.ofFun_mul
@[to_additive (attr := simp)]
lemma UniformOnFun.ofFun_mul [Mul β] (f g : α → β) : ofFun 𝔖 (f * g) = ofFun 𝔖 f * ofFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Mul β] (f g : α → β)
Goal: (UniformOnFun.ofFun 𝔖) (f * g) = (UniformOnFun.ofFun 𝔖) f * (UniformOnFun.ofFun 𝔖) g
-- UniformOnFun.ofFun_add
@[to_additive (attr := simp)]
lemma UniformOnFun.ofFun_mul [Mul β] (f g : α → β) : ofFun 𝔖 (f * g) = ofFun 𝔖 f * ofFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Add β] (f g : α → β)
Goal: (UniformOnFun.ofFun 𝔖) (f + g) = (UniformOnFun.ofFun 𝔖) f + (UniformOnFun.ofFun 𝔖) g
-- UniformFun.toFun_inv
@[to_additive (attr := simp)]
lemma UniformFun.toFun_inv [Inv β] (f : α →ᵤ β) : toFun (f⁻¹) = (toFun f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [Inv β] (f : α →ᵤ β)
Goal: UniformFun.toFun f⁻¹ = (UniformFun.toFun f)⁻¹
-- UniformFun.toFun_neg
@[to_additive (attr := simp)]
lemma UniformFun.toFun_inv [Inv β] (f : α →ᵤ β) : toFun (f⁻¹) = (toFun f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [Neg β] (f : α →ᵤ β)
Goal: UniformFun.toFun (-f) = -UniformFun.toFun f
-- UniformFun.ofFun_neg
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_inv [Inv β] (f : α → β) : ofFun (f⁻¹) = (ofFun f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [Neg β] (f : α → β)
Goal: UniformFun.ofFun (-f) = -UniformFun.ofFun f
-- UniformFun.ofFun_inv
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_inv [Inv β] (f : α → β) : ofFun (f⁻¹) = (ofFun f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [Inv β] (f : α → β)
Goal: UniformFun.ofFun f⁻¹ = (UniformFun.ofFun f)⁻¹
-- UniformOnFun.toFun_inv
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_inv [Inv β] (f : α →ᵤ[𝔖] β) : toFun 𝔖 (f⁻¹) = (toFun 𝔖 f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Inv β] (f : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) f⁻¹ = ((UniformOnFun.toFun 𝔖) f)⁻¹
-- UniformOnFun.toFun_neg
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_inv [Inv β] (f : α →ᵤ[𝔖] β) : toFun 𝔖 (f⁻¹) = (toFun 𝔖 f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Neg β] (f : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) (-f) = -(UniformOnFun.toFun 𝔖) f
-- UniformOnFun.ofFun_neg
@[to_additive (attr := simp)]
lemma UniformOnFun.ofFun_inv [Inv β] (f : α → β) : ofFun 𝔖 (f⁻¹) = (ofFun 𝔖 f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Neg β] (f : α → β)
Goal: (UniformOnFun.ofFun 𝔖) (-f) = -(UniformOnFun.ofFun 𝔖) f
-- UniformOnFun.ofFun_inv
@[to_additive (attr := simp)]
lemma UniformOnFun.ofFun_inv [Inv β] (f : α → β) : ofFun 𝔖 (f⁻¹) = (ofFun 𝔖 f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Inv β] (f : α → β)
Goal: (UniformOnFun.ofFun 𝔖) f⁻¹ = ((UniformOnFun.ofFun 𝔖) f)⁻¹
-- UniformFun.toFun_div
@[to_additive (attr := simp)]
lemma UniformFun.toFun_div [Div β] (f g : α →ᵤ β) : toFun (f / g) = toFun f / toFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Div β] (f g : α →ᵤ β)
Goal: UniformFun.toFun (f / g) = UniformFun.toFun f / UniformFun.toFun g
-- UniformFun.toFun_sub
@[to_additive (attr := simp)]
lemma UniformFun.toFun_div [Div β] (f g : α →ᵤ β) : toFun (f / g) = toFun f / toFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Sub β] (f g : α →ᵤ β)
Goal: UniformFun.toFun (f - g) = UniformFun.toFun f - UniformFun.toFun g
-- UniformFun.ofFun_div
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_div [Div β] (f g : α → β) : ofFun (f / g) = ofFun f / ofFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Div β] (f g : α → β)
Goal: UniformFun.ofFun (f / g) = UniformFun.ofFun f / UniformFun.ofFun g
-- UniformFun.ofFun_sub
@[to_additive (attr := simp)]
lemma UniformFun.ofFun_div [Div β] (f g : α → β) : ofFun (f / g) = ofFun f / ofFun g 
Arguments: {α : Type u_1} {β : Type u_2} [Sub β] (f g : α → β)
Goal: UniformFun.ofFun (f - g) = UniformFun.ofFun f - UniformFun.ofFun g
-- UniformOnFun.toFun_sub
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_div [Div β] (f g : α →ᵤ[𝔖] β) :
    toFun 𝔖 (f / g) = toFun 𝔖 f / toFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Sub β] (f g : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) (f - g) = (UniformOnFun.toFun 𝔖) f - (UniformOnFun.toFun 𝔖) g
-- UniformOnFun.toFun_div
@[to_additive (attr := simp)]
lemma UniformOnFun.toFun_div [Div β] (f g : α →ᵤ[𝔖] β) :
    toFun 𝔖 (f / g) = toFun 𝔖 f / toFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Div β] (f g : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) (f / g) = (UniformOnFun.toFun 𝔖) f / (UniformOnFun.toFun 𝔖) g
-- UniformOnFun.ofFun_sub
@[to_additive (attr := simp)]
lemma UniformOnFun.ofFun_div [Div β] (f g : α → β) : ofFun 𝔖 (f / g) = ofFun 𝔖 f / ofFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Sub β] (f g : α → β)
Goal: (UniformOnFun.ofFun 𝔖) (f - g) = (UniformOnFun.ofFun 𝔖) f - (UniformOnFun.ofFun 𝔖) g
-- UniformOnFun.ofFun_div
@[to_additive (attr := simp)]
lemma UniformOnFun.ofFun_div [Div β] (f g : α → β) : ofFun 𝔖 (f / g) = ofFun 𝔖 f / ofFun 𝔖 g 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [Div β] (f g : α → β)
Goal: (UniformOnFun.ofFun 𝔖) (f / g) = (UniformOnFun.ofFun 𝔖) f / (UniformOnFun.ofFun 𝔖) g
-- UniformFun.toFun_smul
@[simp]
lemma UniformFun.toFun_smul {M : Type*} [SMul M β] (c : M) (f : α →ᵤ β) :
    toFun (c • f) = c • toFun f 
Arguments: {α : Type u_1} {β : Type u_2} {M : Type u_5} [SMul M β] (c : M) (f : α →ᵤ β)
Goal: UniformFun.toFun (c • f) = c • UniformFun.toFun f
-- UniformFun.ofFun_smul
@[simp]
lemma UniformFun.ofFun_smul {M : Type*} [SMul M β] (c : M) (f : α → β) :
    ofFun (c • f) = c • ofFun f 
Arguments: {α : Type u_1} {β : Type u_2} {M : Type u_5} [SMul M β] (c : M) (f : α → β)
Goal: UniformFun.ofFun (c • f) = c • UniformFun.ofFun f
-- UniformOnFun.toFun_smul
@[simp]
lemma UniformOnFun.toFun_smul {M : Type*} [SMul M β] (c : M) (f : α →ᵤ[𝔖] β) :
    toFun 𝔖 (c • f) = c • toFun 𝔖 f 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} {M : Type u_5} [SMul M β] (c : M) (f : α →ᵤ[𝔖] β)
Goal: (UniformOnFun.toFun 𝔖) (c • f) = c • (UniformOnFun.toFun 𝔖) f
-- UniformOnFun.ofFun_smul
@[simp]
lemma UniformOnFun.ofFun_smul {M : Type*} [SMul M β] (c : M) (f : α → β) :
    ofFun 𝔖 (c • f) = c • ofFun 𝔖 f 
Arguments: {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} {M : Type u_5} [SMul M β] (c : M) (f : α → β)
Goal: (UniformOnFun.ofFun 𝔖) (c • f) = c • (UniformOnFun.ofFun 𝔖) f
-- UniformFun.hasBasis_nhds_zero_of_basis
@[to_additive]
protected theorem UniformFun.hasBasis_nhds_one_of_basis {p : ι → Prop} {b : ι → Set G}
    (h : (𝓝 1 : Filter G).HasBasis p b) :
    (𝓝 1 : Filter (α →ᵤ G)).HasBasis p fun i => { f : α →ᵤ G | ∀ x, toFun f x ∈ b i } 
Arguments: {α : Type u_1} {G : Type u_2} {ι : Type u_3} [AddGroup G] [UniformSpace G] [UniformAddGroup G] {p : ι → Prop} {b : ι → Set G} (h : (𝓝 0).HasBasis p b)
Goal: (𝓝 0).HasBasis p fun i => {f | ∀ (x : α), UniformFun.toFun f x ∈ b i}
-- UniformFun.hasBasis_nhds_one_of_basis
@[to_additive]
protected theorem UniformFun.hasBasis_nhds_one_of_basis {p : ι → Prop} {b : ι → Set G}
    (h : (𝓝 1 : Filter G).HasBasis p b) :
    (𝓝 1 : Filter (α →ᵤ G)).HasBasis p fun i => { f : α →ᵤ G | ∀ x, toFun f x ∈ b i } 
Arguments: {α : Type u_1} {G : Type u_2} {ι : Type u_3} [Group G] [UniformSpace G] [UniformGroup G] {p : ι → Prop} {b : ι → Set G} (h : (𝓝 1).HasBasis p b)
Goal: (𝓝 1).HasBasis p fun i => {f | ∀ (x : α), UniformFun.toFun f x ∈ b i}
-- UniformFun.hasBasis_nhds_one
@[to_additive]
protected theorem UniformFun.hasBasis_nhds_one :
    (𝓝 1 : Filter (α →ᵤ G)).HasBasis (fun V : Set G => V ∈ (𝓝 1 : Filter G)) fun V =>
      { f : α → G | ∀ x, f x ∈ V } 
Arguments: {α : Type u_1} {G : Type u_2} [Group G] [UniformSpace G] [UniformGroup G]
Goal: (𝓝 1).HasBasis (fun V => V ∈ 𝓝 1) fun V => {f | ∀ (x : α), f x ∈ V}
-- UniformFun.hasBasis_nhds_zero
@[to_additive]
protected theorem UniformFun.hasBasis_nhds_one :
    (𝓝 1 : Filter (α →ᵤ G)).HasBasis (fun V : Set G => V ∈ (𝓝 1 : Filter G)) fun V =>
      { f : α → G | ∀ x, f x ∈ V } 
Arguments: {α : Type u_1} {G : Type u_2} [AddGroup G] [UniformSpace G] [UniformAddGroup G]
Goal: (𝓝 0).HasBasis (fun V => V ∈ 𝓝 0) fun V => {f | ∀ (x : α), f x ∈ V}