ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.opNorm_nonneg
theorem opNorm_nonneg (f : E →SL[σ₁₂] F) : 0 ≤ ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F)
Goal: 0 ≤ ‖f‖
-- ContinuousLinearMap.opNorm_zero
theorem opNorm_zero : ‖(0 : E →SL[σ₁₂] F)‖ = 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂}
Goal: ‖0‖ = 0
-- ContinuousLinearMap.norm_id_le
theorem norm_id_le : ‖id 𝕜 E‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [SeminormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
Goal: ‖ContinuousLinearMap.id 𝕜 E‖ ≤ 1
-- ContinuousLinearMap.le_opNorm
theorem le_opNorm : ‖f x‖ ≤ ‖f‖ * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E)
Goal: ‖f x‖ ≤ ‖f‖ * ‖x‖
-- ContinuousLinearMap.dist_le_opNorm
theorem dist_le_opNorm (x y : E) : dist (f x) (f y) ≤ ‖f‖ * dist x y 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x y : E)
Goal: dist (f x) (f y) ≤ ‖f‖ * dist x y
-- ContinuousLinearMap.le_of_opNorm_le_of_le
theorem le_of_opNorm_le_of_le {x} {a b : ℝ} (hf : ‖f‖ ≤ a) (hx : ‖x‖ ≤ b) :
    ‖f x‖ ≤ a * b 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {x : E} {a b : ℝ} (hf : ‖f‖ ≤ a) (hx : ‖x‖ ≤ b)
Goal: ‖f x‖ ≤ a * b
-- ContinuousLinearMap.le_opNorm_of_le
theorem le_opNorm_of_le {c : ℝ} {x} (h : ‖x‖ ≤ c) : ‖f x‖ ≤ ‖f‖ * c 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {c : ℝ} {x : E} (h : ‖x‖ ≤ c)
Goal: ‖f x‖ ≤ ‖f‖ * c
-- ContinuousLinearMap.le_of_opNorm_le
theorem le_of_opNorm_le {c : ℝ} (h : ‖f‖ ≤ c) (x : E) : ‖f x‖ ≤ c * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {c : ℝ} (h : ‖f‖ ≤ c) (x : E)
Goal: ‖f x‖ ≤ c * ‖x‖
-- ContinuousLinearMap.opNorm_le_iff
theorem opNorm_le_iff {f : E →SL[σ₁₂] F} {M : ℝ} (hMp : 0 ≤ M) :
    ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {M : ℝ} (hMp : 0 ≤ M)
Goal: ‖f‖ ≤ M ↔ ∀ (x : E), ‖f x‖ ≤ M * ‖x‖
-- ContinuousLinearMap.ratio_le_opNorm
theorem ratio_le_opNorm : ‖f x‖ / ‖x‖ ≤ ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E)
Goal: ‖f x‖ / ‖x‖ ≤ ‖f‖
-- ContinuousLinearMap.unit_le_opNorm
theorem unit_le_opNorm : ‖x‖ ≤ 1 → ‖f x‖ ≤ ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E)
Goal: ‖x‖ ≤ 1 → ‖f x‖ ≤ ‖f‖
-- ContinuousLinearMap.opNorm_le_of_shell
theorem opNorm_le_of_shell {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) {c : 𝕜}
    (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ (x : E), ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f‖ ≤ C
-- ContinuousLinearMap.opNorm_le_of_ball
theorem opNorm_le_of_ball {f : E →SL[σ₁₂] F} {ε : ℝ} {C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C)
    (hf : ∀ x ∈ ball (0 : E) ε, ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) (hf : ∀ x ∈ Metric.ball 0 ε, ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f‖ ≤ C
-- ContinuousLinearMap.opNorm_le_of_nhds_zero
theorem opNorm_le_of_nhds_zero {f : E →SL[σ₁₂] F} {C : ℝ} (hC : 0 ≤ C)
    (hf : ∀ᶠ x in 𝓝 (0 : E), ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {C : ℝ} (hC : 0 ≤ C) (hf : ∀ᶠ (x : E) in 𝓝 0, ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f‖ ≤ C
-- ContinuousLinearMap.opNorm_le_of_shell'
theorem opNorm_le_of_shell' {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) {c : 𝕜}
    (hc : ‖c‖ < 1) (hf : ∀ x, ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) {c : 𝕜} (hc : ‖c‖ < 1) (hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f‖ ≤ C
-- ContinuousLinearMap.opNorm_le_of_unit_norm
theorem opNorm_le_of_unit_norm [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ}
    (hC : 0 ≤ C) (hf : ∀ x, ‖x‖ = 1 → ‖f x‖ ≤ C) : ‖f‖ ≤ C 
Arguments: {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ} (hC : 0 ≤ C) (hf : ∀ (x : E), ‖x‖ = 1 → ‖f x‖ ≤ C)
Goal: ‖f‖ ≤ C
-- ContinuousLinearMap.opNorm_add_le
theorem opNorm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f g : E →SL[σ₁₂] F)
Goal: ‖f + g‖ ≤ ‖f‖ + ‖g‖
-- ContinuousLinearMap.norm_id_of_nontrivial_seminorm
theorem norm_id_of_nontrivial_seminorm (h : ∃ x : E, ‖x‖ ≠ 0) : ‖id 𝕜 E‖ = 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [SeminormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] (h : ∃ x, ‖x‖ ≠ 0)
Goal: ‖ContinuousLinearMap.id 𝕜 E‖ = 1
-- ContinuousLinearMap.opNorm_smul_le
theorem opNorm_smul_le {𝕜' : Type*} [NormedField 𝕜'] [NormedSpace 𝕜' F] [SMulCommClass 𝕜₂ 𝕜' F]
    (c : 𝕜') (f : E →SL[σ₁₂] F) : ‖c • f‖ ≤ ‖c‖ * ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {𝕜' : Type u_11} [NormedField 𝕜'] [NormedSpace 𝕜' F] [SMulCommClass 𝕜₂ 𝕜' F] (c : 𝕜') (f : E →SL[σ₁₂] F)
Goal: ‖c • f‖ ≤ ‖c‖ * ‖f‖
-- _private.Mathlib.Analysis.NormedSpace.OperatorNorm.Basic.0.ContinuousLinearMap.uniformity_eq_seminorm
private lemma uniformity_eq_seminorm :
    𝓤 (E →SL[σ₁₂] F) = ⨅ r > 0, 𝓟 {f | ‖f.1 - f.2‖ < r} 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂]
Goal: 𝓤 (E →SL[σ₁₂] F) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖f.1 - f.2‖ < r}
-- ContinuousLinearMap.opNorm_comp_le
theorem opNorm_comp_le (f : E →SL[σ₁₂] F) : ‖h.comp f‖ ≤ ‖h‖ * ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₃] (h : F →SL[σ₂₃] G) (f : E →SL[σ₁₂] F)
Goal: ‖h.comp f‖ ≤ ‖h‖ * ‖f‖
-- ContinuousLinearMap.opNorm_subsingleton
@[simp, nontriviality]
theorem opNorm_subsingleton [Subsingleton E] : ‖f‖ = 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) [Subsingleton E]
Goal: ‖f‖ = 0
-- ContinuousLinearMap.norm_restrictScalars
@[simp]
theorem norm_restrictScalars (f : E →L[𝕜] Fₗ) : ‖f.restrictScalars 𝕜'‖ = ‖f‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] {𝕜' : Type u_11} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] (f : E →L[𝕜] Fₗ)
Goal: ‖ContinuousLinearMap.restrictScalars 𝕜' f‖ = ‖f‖
-- ContinuousLinearMap.coe_restrictScalarsIsometry
@[simp]
theorem coe_restrictScalarsIsometry :
    ⇑(restrictScalarsIsometry 𝕜 E Fₗ 𝕜' 𝕜'') = restrictScalars 𝕜' 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (𝕜' : Type u_11) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ]
Goal: ⇑(ContinuousLinearMap.restrictScalarsIsometry 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalars 𝕜'
-- ContinuousLinearMap.restrictScalarsIsometry_toLinearMap
@[simp]
theorem restrictScalarsIsometry_toLinearMap :
    (restrictScalarsIsometry 𝕜 E Fₗ 𝕜' 𝕜'').toLinearMap = restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜'' 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (𝕜' : Type u_11) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ]
Goal: (ContinuousLinearMap.restrictScalarsIsometry 𝕜 E Fₗ 𝕜' 𝕜'').toLinearMap =
  ContinuousLinearMap.restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜''
-- ContinuousLinearMap.coe_restrictScalarsL
@[simp]
theorem coe_restrictScalarsL : (restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'' : (E →L[𝕜] Fₗ) →ₗ[𝕜''] E →L[𝕜'] Fₗ) =
    restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜'' 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] {𝕜' : Type u_11} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ]
Goal: ↑(ContinuousLinearMap.restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜''
-- ContinuousLinearMap.coe_restrict_scalarsL'
@[simp]
theorem coe_restrict_scalarsL' : ⇑(restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = restrictScalars 𝕜' 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] {𝕜' : Type u_11} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ]
Goal: ⇑(ContinuousLinearMap.restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalars 𝕜'
-- ContinuousLinearMap.norm_pi_le_of_le
lemma norm_pi_le_of_le {ι : Type*} [Fintype ι]
    {M : ι → Type*} [∀ i, SeminormedAddCommGroup (M i)] [∀ i, NormedSpace 𝕜 (M i)] {C : ℝ}
    {L : (i : ι) → (E →L[𝕜] M i)} (hL : ∀ i, ‖L i‖ ≤ C) (hC : 0 ≤ C) :
    ‖pi L‖ ≤ C 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [SeminormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {ι : Type u_11} [Fintype ι] {M : ι → Type u_12} [(i : ι) → SeminormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {C : ℝ} {L : (i : ι) → E →L[𝕜] M i} (hL : ∀ (i : ι), ‖L i‖ ≤ C) (hC : 0 ≤ C)
Goal: ‖ContinuousLinearMap.pi L‖ ≤ C
-- LinearMap.mkContinuous_norm_le
theorem mkContinuous_norm_le (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (hC : 0 ≤ C) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    ‖f.mkContinuous C h‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (hC : 0 ≤ C) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f.mkContinuous C h‖ ≤ C
-- LinearMap.mkContinuous_norm_le'
theorem mkContinuous_norm_le' (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    ‖f.mkContinuous C h‖ ≤ max C 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f.mkContinuous C h‖ ≤ max C 0
-- LinearIsometry.norm_toContinuousLinearMap_le
theorem norm_toContinuousLinearMap_le (f : E →ₛₗᵢ[σ₁₂] F) : ‖f.toContinuousLinearMap‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗᵢ[σ₁₂] F)
Goal: ‖f.toContinuousLinearMap‖ ≤ 1
-- Submodule.norm_subtypeL_le
theorem norm_subtypeL_le (K : Submodule 𝕜 E) : ‖K.subtypeL‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [SeminormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] (K : Submodule 𝕜 E)
Goal: ‖K.subtypeL‖ ≤ 1
-- ne_neg_of_mem_sphere
theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [CharZero 𝕜] {r : ℝ} (hr : r ≠ 0) (x : ↑(Metric.sphere 0 r))
Goal: x ≠ -x
-- ne_neg_of_mem_unit_sphere
theorem ne_neg_of_mem_unit_sphere (x : sphere (0 : E) 1) : x ≠ -x 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [CharZero 𝕜] (x : ↑(Metric.sphere 0 1))
Goal: x ≠ -x
-- balancedCore_subset
theorem balancedCore_subset (s : Set E) : balancedCore 𝕜 s ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] (s : Set E)
Goal: balancedCore 𝕜 s ⊆ s
-- balancedCore_empty
theorem balancedCore_empty : balancedCore 𝕜 (∅ : Set E) = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E]
Goal: balancedCore 𝕜 ∅ = ∅
-- mem_balancedCore_iff
theorem mem_balancedCore_iff : x ∈ balancedCore 𝕜 s ↔ ∃ t, Balanced 𝕜 t ∧ t ⊆ s ∧ x ∈ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] {s : Set E} {x : E}
Goal: x ∈ balancedCore 𝕜 s ↔ ∃ t, Balanced 𝕜 t ∧ t ⊆ s ∧ x ∈ t
-- smul_balancedCore_subset
theorem smul_balancedCore_subset (s : Set E) {a : 𝕜} (ha : ‖a‖ ≤ 1) :
    a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] (s : Set E) {a : 𝕜} (ha : ‖a‖ ≤ 1)
Goal: a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s
-- balancedCore_balanced
theorem balancedCore_balanced (s : Set E) : Balanced 𝕜 (balancedCore 𝕜 s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] (s : Set E)
Goal: Balanced 𝕜 (balancedCore 𝕜 s)
-- Balanced.subset_balancedCore_of_subset
theorem Balanced.subset_balancedCore_of_subset (hs : Balanced 𝕜 s) (h : s ⊆ t) :
    s ⊆ balancedCore 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] {s t : Set E} (hs : Balanced 𝕜 s) (h : s ⊆ t)
Goal: s ⊆ balancedCore 𝕜 t
-- mem_balancedCoreAux_iff
theorem mem_balancedCoreAux_iff : x ∈ balancedCoreAux 𝕜 s ↔ ∀ r : 𝕜, 1 ≤ ‖r‖ → x ∈ r • s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] {s : Set E} {x : E}
Goal: x ∈ balancedCoreAux 𝕜 s ↔ ∀ (r : 𝕜), 1 ≤ ‖r‖ → x ∈ r • s
-- mem_balancedHull_iff
theorem mem_balancedHull_iff : x ∈ balancedHull 𝕜 s ↔ ∃ r : 𝕜, ‖r‖ ≤ 1 ∧ x ∈ r • s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] {s : Set E} {x : E}
Goal: x ∈ balancedHull 𝕜 s ↔ ∃ r, ‖r‖ ≤ 1 ∧ x ∈ r • s
-- Balanced.balancedHull_subset_of_subset
theorem Balanced.balancedHull_subset_of_subset (ht : Balanced 𝕜 t) (h : s ⊆ t) :
    balancedHull 𝕜 s ⊆ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [SMul 𝕜 E] {s t : Set E} (ht : Balanced 𝕜 t) (h : s ⊆ t)
Goal: balancedHull 𝕜 s ⊆ t
-- balancedCore_zero_mem
theorem balancedCore_zero_mem (hs : (0 : E) ∈ s) : (0 : E) ∈ balancedCore 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : 0 ∈ s)
Goal: 0 ∈ balancedCore 𝕜 s
-- balancedCore_nonempty_iff
theorem balancedCore_nonempty_iff : (balancedCore 𝕜 s).Nonempty ↔ (0 : E) ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: (balancedCore 𝕜 s).Nonempty ↔ 0 ∈ s
-- subset_balancedHull
theorem subset_balancedHull [NormOneClass 𝕜] {s : Set E} : s ⊆ balancedHull 𝕜 s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormOneClass 𝕜] {s : Set E}
Goal: s ⊆ balancedHull 𝕜 s
-- balancedHull.balanced
theorem balancedHull.balanced (s : Set E) : Balanced 𝕜 (balancedHull 𝕜 s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: Balanced 𝕜 (balancedHull 𝕜 s)
-- balancedCoreAux_empty
@[simp]
theorem balancedCoreAux_empty : balancedCoreAux 𝕜 (∅ : Set E) = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
Goal: balancedCoreAux 𝕜 ∅ = ∅
-- balancedCoreAux_subset
theorem balancedCoreAux_subset (s : Set E) : balancedCoreAux 𝕜 s ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: balancedCoreAux 𝕜 s ⊆ s
-- balancedCoreAux_balanced
theorem balancedCoreAux_balanced (h0 : (0 : E) ∈ balancedCoreAux 𝕜 s) :
    Balanced 𝕜 (balancedCoreAux 𝕜 s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h0 : 0 ∈ balancedCoreAux 𝕜 s)
Goal: Balanced 𝕜 (balancedCoreAux 𝕜 s)
-- balancedCoreAux_maximal
theorem balancedCoreAux_maximal (h : t ⊆ s) (ht : Balanced 𝕜 t) : t ⊆ balancedCoreAux 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (h : t ⊆ s) (ht : Balanced 𝕜 t)
Goal: t ⊆ balancedCoreAux 𝕜 s
-- balancedCore_subset_balancedCoreAux
theorem balancedCore_subset_balancedCoreAux : balancedCore 𝕜 s ⊆ balancedCoreAux 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: balancedCore 𝕜 s ⊆ balancedCoreAux 𝕜 s
-- balancedCore_eq_iInter
theorem balancedCore_eq_iInter (hs : (0 : E) ∈ s) :
    balancedCore 𝕜 s = ⋂ (r : 𝕜) (_ : 1 ≤ ‖r‖), r • s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : 0 ∈ s)
Goal: balancedCore 𝕜 s = ⋂ r, ⋂ (_ : 1 ≤ ‖r‖), r • s
-- subset_balancedCore
theorem subset_balancedCore (ht : (0 : E) ∈ t) (hst : ∀ a : 𝕜, ‖a‖ ≤ 1 → a • s ⊆ t) :
    s ⊆ balancedCore 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (ht : 0 ∈ t) (hst : ∀ (a : 𝕜), ‖a‖ ≤ 1 → a • s ⊆ t)
Goal: s ⊆ balancedCore 𝕜 t
-- IsClosed.balancedCore
protected theorem IsClosed.balancedCore (hU : IsClosed U) : IsClosed (balancedCore 𝕜 U) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] {U : Set E} (hU : IsClosed U)
Goal: IsClosed (balancedCore 𝕜 U)
-- balancedCore_mem_nhds_zero
theorem balancedCore_mem_nhds_zero (hU : U ∈ 𝓝 (0 : E)) : balancedCore 𝕜 U ∈ 𝓝 (0 : E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] {U : Set E} (hU : U ∈ 𝓝 0)
Goal: balancedCore 𝕜 U ∈ 𝓝 0
-- nhds_basis_balanced
theorem nhds_basis_balanced :
    (𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ 𝓝 (0 : E) ∧ Balanced 𝕜 s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E]
Goal: (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Balanced 𝕜 s) id
-- nhds_basis_closed_balanced
theorem nhds_basis_closed_balanced [RegularSpace E] :
    (𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ 𝓝 (0 : E) ∧ IsClosed s ∧ Balanced 𝕜 s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [RegularSpace E]
Goal: (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ IsClosed s ∧ Balanced 𝕜 s) id
-- nhds_basis_abs_convex
theorem nhds_basis_abs_convex :
    (𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ 𝓝 (0 : E) ∧ Balanced 𝕜 s ∧ Convex ℝ s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Module ℝ E] [SMulCommClass ℝ 𝕜 E] [TopologicalSpace E] [LocallyConvexSpace ℝ E] [ContinuousSMul 𝕜 E]
Goal: (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Balanced 𝕜 s ∧ Convex ℝ s) id
-- nhds_basis_abs_convex_open
theorem nhds_basis_abs_convex_open :
    (𝓝 (0 : E)).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Module ℝ E] [SMulCommClass ℝ 𝕜 E] [TopologicalSpace E] [LocallyConvexSpace ℝ E] [ContinuousSMul 𝕜 E] [ContinuousSMul ℝ E] [TopologicalAddGroup E]
Goal: (𝓝 0).HasBasis (fun s => 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s) id
-- AbsConvexOpenSets.coe_zero_mem
theorem coe_zero_mem (s : AbsConvexOpenSets 𝕜 E) : (0 : E) ∈ (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace E] [AddCommMonoid E] [Zero E] [SeminormedRing 𝕜] [SMul 𝕜 E] [SMul ℝ E] (s : AbsConvexOpenSets 𝕜 E)
Goal: 0 ∈ ↑s
-- AbsConvexOpenSets.coe_isOpen
theorem coe_isOpen (s : AbsConvexOpenSets 𝕜 E) : IsOpen (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace E] [AddCommMonoid E] [Zero E] [SeminormedRing 𝕜] [SMul 𝕜 E] [SMul ℝ E] (s : AbsConvexOpenSets 𝕜 E)
Goal: IsOpen ↑s