mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
hcont : Continuous[Pi.topologicalSpace, inst✝] ((toFun {s | IsVonNBounded 𝕜 s}) f)
hadd :
  ∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
      (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)
hsmul :
  ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) = c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)
⊢ ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), f (update m i (c • x)) = c • f (update m i x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
⊢ range toUniformOnFun =
    {f |
      Continuous[Pi.topologicalSpace, inst✝] ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
        (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
            (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
          ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
            (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
              c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
⊢ (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : UniformSpace F
inst✝ : IsUniformAddGroup F
⊢ instTopologicalSpace =
    (UniformSpace.comap toUniformOnFun
        (UniformOnFun.uniformSpace ((i : ι) → E i) F {s | IsVonNBounded 𝕜 s})).toTopologicalSpace

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : IsUniformAddGroup F
G : Type u_5
inst✝³ : AddCommGroup G
inst✝² : UniformSpace G
inst✝¹ : IsUniformAddGroup G
inst✝ : Module 𝕜 G
g : F →L[𝕜] G
hg : IsUniformInducing ⇑g
⊢ IsUniformInducing g.compContinuousMultilinearMap

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
⊢ (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
⊢ ∀ (x : E), ‖0 x‖ ≤ 0 * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : Semiring 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
⊢ ∀ (x : E), ‖LinearMap.id x‖ ≤ 1 * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
⊢ IsBoundedLinearMap 𝕜 fun x ↦ x.1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
⊢ IsBoundedLinearMap 𝕜 fun x ↦ x.2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : Semiring 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : Module 𝕜 F
f : E → F
𝕜' : Type u_5
c : 𝕜'
inst✝³ : SeminormedRing 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : SMulCommClass 𝕜 𝕜' F
hf : IsBoundedLinearMap 𝕜 f
hlf : IsLinearMap 𝕜 f
M : ℝ
left✝ : 0 < M
hM : ∀ (x : E), ‖f x‖ ≤ M * ‖x‖
x : E
⊢ ‖c‖ * ‖f x‖ ≤ ‖c‖ * (M * ‖x‖)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E (SeparationQuotient F))

pp: case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ Surjective (SeparationQuotient.mkCLM 𝕜 F).compContinuousMultilinearMap

Tactic:
· exact this inferInstance

Goal state after:
case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ Surjective (SeparationQuotient.mkCLM 𝕜 F).compContinuousMultilinearMap
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
f : E → F
hf : IsBoundedLinearMap 𝕜 f
⊢ ∃ M, 0 < M ∧ ∀ (x : E), ‖-f x‖ ≤ M * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
f g : E → F
hf : IsBoundedLinearMap 𝕜 f
hg : IsBoundedLinearMap 𝕜 g
hlf : IsLinearMap 𝕜 f
Mf : ℝ
left✝¹ : 0 < Mf
hMf : ∀ (x : E), ‖f x‖ ≤ Mf * ‖x‖
hlg : IsLinearMap 𝕜 g
Mg : ℝ
left✝ : 0 < Mg
hMg : ∀ (x : E), ‖g x‖ ≤ Mg * ‖x‖
x : E
⊢ Mf * ‖x‖ + Mg * ‖x‖ ≤ (Mf + Mg) * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
f g : E → F
hf : IsBoundedLinearMap 𝕜 f
hg : IsBoundedLinearMap 𝕜 g
⊢ IsBoundedLinearMap 𝕜 fun e ↦ f e - g e

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
⊢ (∃ C, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
C : ℝ
hC_pos : C > 0
hC : ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
x : F
y : E
⊢ C * ‖y‖ * ‖x‖ = C * ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
⊢ ∃ C > 0, ∀ (x : F) (y : E), ‖f ((x, y).2, (x, y).1)‖ ≤ C * ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E
C : ℝ
hC_pos : C > 0
hC : ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
⊢ 0 < C * max ‖x‖ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E
⊢ ∃ M, 0 < M ∧ ∀ (x_1 : F), ‖f (x, x_1)‖ ≤ M * ‖x_1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_6_iff_8 : (∃ C, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
⊢ BddAbove (Set.range fun x ↦ ‖f x‖) ↔ ⨆ i, ↑‖f i‖₊ < ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
C : ℝ
left✝ : C > 0
hC : ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
x✝ : E × F
x : E
y : F
⊢ ‖f (x, y)‖ ≤ C * ‖‖(x, y).1‖ * ‖(x, y).2‖‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
this : Tendsto (fun y ↦ f (y.1 - x.1, y.2) + f (x.1, y.2 - x.2)) (𝓝 x) (𝓝 (0 + 0))
⊢ Tendsto (fun x_1 ↦ f x_1 - f x) (𝓝 x) (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
⊢ Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
e₂ : F
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, instTopologicalSpaceProd] fun e₁ ↦ (e₁, e₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
e₁ : E
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, instTopologicalSpaceProd] (Prod.mk e₁)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : SeminormedRing A
inst✝⁵ : Algebra 𝕜 A
E : Type u_3
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module A E
inst✝¹ : IsBoundedSMul A E
inst✝ : IsScalarTower 𝕜 A E
x : A
y : E
⊢ ‖(x, y).1 • (x, y).2‖ ≤ 1 * ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E →L[𝕜] F →L[𝕜] G
x : E
y : F
⊢ ‖f‖ * ‖x‖ * ‖y‖ ≤ max ‖f‖ 1 * ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
X : Type u_5
inst✝ : TopologicalSpace X
f : X → E →L[𝕜] G
g : X → F →L[𝕜] G
s : Set X
x : X
hf : ContinuousWithinAt f s x
hg : ContinuousWithinAt g s x
⊢ ContinuousWithinAt (fun x ↦ (f x).coprod (g x)) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
X : Type u_5
inst✝ : TopologicalSpace X
f : X → E →L[𝕜] G
g : X → F →L[𝕜] G
x : X
hf : ContinuousAt f x
hg : ContinuousAt g x
⊢ ContinuousAt (fun x ↦ (f x).coprod (g x)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
X : Type u_5
inst✝ : TopologicalSpace X
f : X → E →L[𝕜] G
g : X → F →L[𝕜] G
s : Set X
hf : ContinuousOn f s
hg : ContinuousOn g s
⊢ ContinuousOn (fun x ↦ (f x).coprod (g x)) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_6_iff_8 : (∃ C, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_8_iff_9 : BddAbove (Set.range fun x ↦ ‖f x‖) ↔ ⨆ i, ↑‖f i‖₊ < ⊤
⊢ UniformEquicontinuous (DFunLike.coe ∘ f) ↔ ∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
X : Type u_5
inst✝ : TopologicalSpace X
f : X → E →L[𝕜] G
g : X → F →L[𝕜] G
hf : Continuous[inst✝, _] f
hg : Continuous[inst✝, _] g
⊢ Continuous[inst✝, topologicalSpace] fun x ↦ (f x).coprod (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
⊢ [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
      ∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖, ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖,
      ∃ C, ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (Set.range fun x ↦ ‖f x‖),
      ⨆ i, ↑‖f i‖₊ < ⊤].TFAE

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Determinant

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : CommRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : ContinuousMul 𝕜
f : 𝕜 →L[𝕜] 𝕜
v : 𝕜
⊢ (f.smulRight v).det = f 1 * v

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Determinant

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : CommRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : ContinuousMul 𝕜
v : 𝕜
⊢ (toSpanSingleton 𝕜 v).det = v

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Simple

Goal state before:
pp: R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
l : M →ₗ[R] N
⊢ IsClosed ↑l.ker ∨ Dense ↑l.ker

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : M →L[R] M
⊢ IsIdempotentElem ↑f ↔ IsIdempotentElem f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p q : M →L[R] M
hp : IsIdempotentElem p
hq : IsIdempotentElem q
⊢ p = q ↔ (↑p).range = (↑q).range ∧ (↑p).ker = (↑q).ker

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f T : M →L[R] M
hf : IsIdempotentElem f
⊢ (↑f).range ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = T ∘SL f

Tactic:
by

Goal state after: