mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
ts : Set (TopologicalSpace E)
hts : ∀ t ∈ ts, PolynormableSpace 𝕜 E
⊢ PolynormableSpace 𝕜 E

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f✝ : l.Germ M
f : α → M
⊢ 0 • ↑f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t₁ t₂ : TopologicalSpace E
ht₁ : PolynormableSpace 𝕜 E
ht₂ : PolynormableSpace 𝕜 E
⊢ ∀ t ∈ {t₁, t₂}, PolynormableSpace 𝕜 E

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t₁ t₂ : TopologicalSpace E
ht₁ : PolynormableSpace 𝕜 E
ht₂ : PolynormableSpace 𝕜 E
⊢ PolynormableSpace 𝕜 E

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
c₁✝ c₂✝ : l.Germ R
f✝ : l.Germ M
c₁ c₂ : α → R
f : α → M
⊢ (↑c₁ + ↑c₂) • ↑f = ↑c₁ • ↑f + ↑c₂ • ↑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Countable ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
this✝ : IsTopologicalAddGroup E
x✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
⊢ (𝓝 0).IsCountablyGenerated

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Countable ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ FirstCountableTopology E

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝ g✝ h✝ : α → β
inst✝² : Mul β
inst✝¹ : LE β
inst✝ : ExistsMulOfLE β
x y : l.Germ β
f g : α → β
h : f ≤ᶠ[l] g
⊢ ∃ c, ↑g = ↑f * c

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactlyCoherentSpace X
s : Set X
⊢ IsOpen[inst✝¹] s ↔
    ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X), Continuous[_, inst✝¹] f → IsOpen (f ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
h :
  ∀ (s : Set X),
    (∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X),
        Continuous[_, inst✝] f → IsOpen (f ⁻¹' s)) →
      IsOpen[inst✝] s
⊢ CompactlyCoherentSpace X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
⊢ CompactCoherentification.map f ∘ ⇑(CompactCoherentification.mk X) = ⇑(CompactCoherentification.mk Y) ∘ f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set (𝐤X)
⊢ IsOpen[instTopologicalSpace] A ↔
    ∀ (K : Set X), IsCompact K → IsOpen[instTopologicalSpaceSubtype] (K ↓∩ ⇑(CompactCoherentification.mk X) ⁻¹' A)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set (𝐤X)
⊢ IsClosed[instTopologicalSpace] A ↔
    ∀ (K : Set X), IsCompact K → IsClosed[instTopologicalSpaceSubtype] (K ↓∩ ⇑(CompactCoherentification.mk X) ⁻¹' A)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : 𝐤X → Y
⊢ Continuous[instTopologicalSpace, inst✝] f ↔
    ∀ (K : Set X), IsCompact K → ContinuousOn (f ∘ ⇑(CompactCoherentification.mk X)) K

Tactic:
simp_rw [continuous_coinduced_dom, continuous_iSup_dom, continuous_coinduced_dom,
    continuousOn_iff_continuous_domRestrict]
  rfl

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : 𝐤X → Y
⊢ Continuous[instTopologicalSpace, inst✝] f ↔
    ∀ (K : Set X), IsCompact K → ContinuousOn (f ∘ ⇑(CompactCoherentification.mk X)) K

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ Continuous[instTopologicalSpace, inst✝] ⇑(CompactCoherentification.mk X).symm

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ IsOpenMap ⇑(CompactCoherentification.mk X)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : Set X
hK : IsCompact K
⊢ ContinuousOn (⇑(CompactCoherentification.mk X)) K

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → 𝐤Y
K : Set X
hK : IsCompact K
⊢ ContinuousOn f K ↔ ContinuousOn (⇑(CompactCoherentification.mk Y).symm ∘ f) K

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactlyCoherentSpace X
⊢ Continuous[inst✝¹, instTopologicalSpace] ⇑(CompactCoherentification.mk X)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → 𝐤Y
inst✝ : CompactSpace X
⊢ Continuous[inst✝², instTopologicalSpace] f ↔ Continuous[inst✝², inst✝¹] (⇑(CompactCoherentification.mk Y).symm ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : ∀ (K : Set X), IsCompact K → ContinuousOn f K
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] (CompactCoherentification.map f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] (CompactCoherentification.map f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : CompactSpace X
f : X → Y
⊢ Continuous[inst✝², instTopologicalSpace] (⇑(CompactCoherentification.mk Y) ∘ f) ↔ Continuous[inst✝², inst✝¹] f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
A : Set X
hA : IsCompact A
f : X → Y
⊢ ContinuousOn (⇑(CompactCoherentification.mk Y) ∘ f) A ↔ ContinuousOn f A

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : Set (𝐤X)
⊢ IsCompact K ↔ IsCompact (⇑(CompactCoherentification.mk X) ⁻¹' K)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : Set X
⊢ IsCompact (⇑(CompactCoherentification.mk X) '' K) ↔ IsCompact K

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ CompactlyCoherentSpace (𝐤X)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : SeminormedAddCommGroup F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₂] F
g : E →SL[σ₁₃] G
h : ∀ (x : E), ‖f x‖ = ‖g x‖
x : E
⊢ ‖f x‖ ≤ ‖g‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁵ : NormedField 𝕜₁
inst✝¹⁴ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E✝ : Type u_3
F✝ : Type u_4
G : Type u_5
inst✝¹³ : AddCommGroup E✝
inst✝¹² : Module 𝕜₁ E✝
inst✝¹¹ : TopologicalSpace E✝
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜₂ F✝
𝕜 : Type u_6
inst✝⁸ : NormedField 𝕜
E : Type u_7
ι : Type u_8
F : ι → Type u_9
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : (i : ι) → AddCommGroup (F i)
inst✝³ : (i : ι) → Module 𝕜 (F i)
inst✝² : (i : ι) → TopologicalSpace (F i)
inst✝¹ : ∀ (i : ι), IsTopologicalAddGroup (F i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (F i)
𝔖 : Set (Set E)
this✝ : (i : ι) → UniformSpace (F i) := fun i ↦ IsTopologicalAddGroup.rightUniformSpace (F i)
this : ∀ (i : ι), IsUniformAddGroup (F i)
⊢ Continuous[Pi.topologicalSpace, instTopologicalSpace (RingHom.id 𝕜) ((i : ι) → F i) 𝔖] fun F_1 ↦
    ContinuousLinearMap.pi F_1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : SeminormedAddCommGroup F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₂] F
g : E →SL[σ₁₃] G
h : ∀ (x : E), ‖f x‖ = ‖g x‖
c : ℝ
hc : c ≥ 0
h₂ : ∀ (x : E), ‖f x‖ ≤ c * ‖x‖
z : E
⊢ ‖g z‖ ≤ c * ‖z‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : SeminormedAddCommGroup F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝ : RingHomIsometric σ₂₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
C : ℝ
h0 : 0 ≤ C
hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖
x : E
⊢ 0 ≤ C * ‖x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup F
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NontriviallyNormedField 𝕜₃
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
inst✝ : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G
C : ℝ
h : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖
x : E
⊢ max (C * ‖x‖) 0 = max C 0 * ‖x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁷ : SeminormedAddCommGroup E
inst✝¹⁶ : SeminormedAddCommGroup Eₗ
inst✝¹⁵ : SeminormedAddCommGroup F
inst✝¹⁴ : SeminormedAddCommGroup Fₗ
inst✝¹³ : SeminormedAddCommGroup G
inst✝¹² : SeminormedAddCommGroup Gₗ
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜₂
inst✝⁹ : NontriviallyNormedField 𝕜₃
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedSpace 𝕜 Eₗ
inst✝⁶ : NormedSpace 𝕜₂ F
inst✝⁵ : NormedSpace 𝕜 Fₗ
inst✝⁴ : NormedSpace 𝕜₃ G
inst✝³ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : FunLike 𝓕 E F
inst✝ : RingHomIsometric σ₂₃
f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G
h : ∃ C, ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖
x y : E
⊢ (f (x + y)).mkContinuousOfExistsBound ⋯ = (f x).mkContinuousOfExistsBound ⋯ + (f y).mkContinuousOfExistsBound ⋯

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁷ : SeminormedAddCommGroup E
inst✝¹⁶ : SeminormedAddCommGroup Eₗ
inst✝¹⁵ : SeminormedAddCommGroup F
inst✝¹⁴ : SeminormedAddCommGroup Fₗ
inst✝¹³ : SeminormedAddCommGroup G
inst✝¹² : SeminormedAddCommGroup Gₗ
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜₂
inst✝⁹ : NontriviallyNormedField 𝕜₃
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedSpace 𝕜 Eₗ
inst✝⁶ : NormedSpace 𝕜₂ F
inst✝⁵ : NormedSpace 𝕜 Fₗ
inst✝⁴ : NormedSpace 𝕜₃ G
inst✝³ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : FunLike 𝓕 E F
inst✝ : RingHomIsometric σ₂₃
f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G
h : ∃ C, ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖
c : 𝕜
x : E
⊢ (f (c • x)).mkContinuousOfExistsBound ⋯ = σ₁₃ c • (f x).mkContinuousOfExistsBound ⋯

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁸ : SeminormedAddCommGroup E
inst✝¹⁷ : SeminormedAddCommGroup Eₗ
inst✝¹⁶ : SeminormedAddCommGroup F
inst✝¹⁵ : SeminormedAddCommGroup Fₗ
inst✝¹⁴ : SeminormedAddCommGroup G
inst✝¹³ : SeminormedAddCommGroup Gₗ
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NontriviallyNormedField 𝕜₂
inst✝¹⁰ : NontriviallyNormedField 𝕜₃
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 Eₗ
inst✝⁷ : NormedSpace 𝕜₂ F
inst✝⁶ : NormedSpace 𝕜 Fₗ
inst✝⁵ : NormedSpace 𝕜₃ G
inst✝⁴ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝² : FunLike 𝓕 E F
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
z : F
x y : E
⊢ (f (x + y)) z = (f x) z + (f y) z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁸ : SeminormedAddCommGroup E
inst✝¹⁷ : SeminormedAddCommGroup Eₗ
inst✝¹⁶ : SeminormedAddCommGroup F
inst✝¹⁵ : SeminormedAddCommGroup Fₗ
inst✝¹⁴ : SeminormedAddCommGroup G
inst✝¹³ : SeminormedAddCommGroup Gₗ
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NontriviallyNormedField 𝕜₂
inst✝¹⁰ : NontriviallyNormedField 𝕜₃
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 Eₗ
inst✝⁷ : NormedSpace 𝕜₂ F
inst✝⁶ : NormedSpace 𝕜 Fₗ
inst✝⁵ : NormedSpace 𝕜₃ G
inst✝⁴ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝² : FunLike 𝓕 E F
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
c : 𝕜
y : F
x : E
⊢ (f (c • x)) y = σ₁₃ c • (f x) y

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁸ : SeminormedAddCommGroup E
inst✝¹⁷ : SeminormedAddCommGroup Eₗ
inst✝¹⁶ : SeminormedAddCommGroup F
inst✝¹⁵ : SeminormedAddCommGroup Fₗ
inst✝¹⁴ : SeminormedAddCommGroup G
inst✝¹³ : SeminormedAddCommGroup Gₗ
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NontriviallyNormedField 𝕜₂
inst✝¹⁰ : NontriviallyNormedField 𝕜₃
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 Eₗ
inst✝⁷ : NormedSpace 𝕜₂ F
inst✝⁶ : NormedSpace 𝕜 Fₗ
inst✝⁵ : NormedSpace 𝕜₃ G
inst✝⁴ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝² : FunLike 𝓕 E F
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
y : F
x : E
⊢ ‖f‖ * ‖x‖ * ‖y‖ = ‖f‖ * ‖y‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ Tendsto F p (𝓝 f) ↔ ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
x : E
y : F
⊢ ‖(f x) y‖ ≤ ‖f.flip‖ * ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
⊢ compactOpen = (UniformSpace.comap toUniformOnFunIsCompact inferInstance).toTopologicalSpace

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u_1
pi : ι → Prop
s : ι → Set (β × β)
h : (𝓤 β).HasBasis pi s
⊢ (𝓤 C(α, β)).HasBasis (fun p ↦ IsCompact p.1 ∧ pi p.2) fun p ↦ {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2}

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
⊢ f.flip.flip = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
X : Set (C(α, β) × C(α, β))
⊢ X ∈ 𝓤 C(α, β) ↔ ∃ K V, IsCompact K ∧ V ∈ 𝓤 β ∧ {fg | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
⊢ ‖f.flip‖ ≤ ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
h : TendstoLocallyUniformly (fun i a ↦ (F i) a) (⇑f) p
⊢ Tendsto F p (𝓝 f)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
⊢ ‖f.flip‖₊ = ‖f‖₊

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
⊢ ‖f.flip‖ₑ = ‖f‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : WeaklyLocallyCompactSpace α
⊢ Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a ↦ (F i) a) (⇑f) p

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁹ : SeminormedAddCommGroup E
inst✝¹⁸ : SeminormedAddCommGroup Eₗ
inst✝¹⁷ : SeminormedAddCommGroup F
inst✝¹⁶ : SeminormedAddCommGroup Fₗ
inst✝¹⁵ : SeminormedAddCommGroup G
inst✝¹⁴ : SeminormedAddCommGroup Gₗ
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NontriviallyNormedField 𝕜₂
inst✝¹¹ : NontriviallyNormedField 𝕜₃
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 Eₗ
inst✝⁸ : NormedSpace 𝕜₂ F
inst✝⁷ : NormedSpace 𝕜 Fₗ
inst✝⁶ : NormedSpace 𝕜₃ G
inst✝⁵ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝³ : FunLike 𝓕 E F
inst✝² : RingHomIsometric σ₂₃
inst✝¹ : RingHomIsometric σ₁₃
inst✝ : RingHomIsometric σ₁₂
c : 𝕜₂
f : F →SL[σ₂₃] G
g : E →SL[σ₁₂] F
⊢ f ∘SL (c • g) = σ₂₃ c • f ∘SL g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : CompactSpace α
⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a ↦ (F i) a) (⇑f) p

Tactic:
by

Goal state after: