mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case refine_3
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
X : Type u_6
inst✝ : TopologicalSpace X
f : TotalSpace F E → X
s : Set B
hs : IsOpen[inst✝³] s
hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ univ)
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
z : TotalSpace F E
hz : z ∈ TotalSpace.proj ⁻¹' s
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ z.proj ∈ s ∩ (a.pretrivializationAt z.proj).baseSet

pp: case refine_3
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
X : Type u_6
inst✝ : TopologicalSpace X
f : TotalSpace F E → X
s : Set B
hs : IsOpen[inst✝³] s
hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ univ)
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
z : TotalSpace F E
hz : z ∈ TotalSpace.proj ⁻¹' s
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ z ∈ e.source

Tactic:
]

Goal state after:
case refine_3
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
X : Type u_6
inst✝ : TopologicalSpace X
f : TotalSpace F E → X
s : Set B
hs : IsOpen[inst✝³] s
hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ univ)
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
z : TotalSpace F E
hz : z ∈ TotalSpace.proj ⁻¹' s
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ z.proj ∈ s ∩ (a.pretrivializationAt z.proj).baseSet

case refine_3
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
X : Type u_6
inst✝ : TopologicalSpace X
f : TotalSpace F E → X
s : Set B
hs : IsOpen[inst✝³] s
hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ univ)
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
z : TotalSpace F E
hz : z ∈ TotalSpace.proj ⁻¹' s
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ z ∈ e.source
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalSpace (TotalSpace F E)
inst✝ : ∀ (x : B), Nonempty (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) e.target

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalSpace (TotalSpace F E)
inst✝ : ∀ (x : B), Nonempty (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) e.target
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ (b, e₁.coordChange e₂ b x).1 = (b, x).1

pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ proj (↑e₁.symm (b, x)) ∈ e₁.baseSet

pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ proj (↑e₁.symm (b, x)) ∈ e₂.baseSet

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ proj (↑e₁.symm (b, x)) ∈ e₁.baseSet

B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ proj (↑e₁.symm (b, x)) ∈ e₂.baseSet
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case refine_3
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
X : Type u_6
inst✝ : TopologicalSpace X
f : TotalSpace F E → X
s : Set B
hs : IsOpen[inst✝³] s
hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ univ)
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
z : TotalSpace F E
hz : z ∈ TotalSpace.proj ⁻¹' s
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ z.proj ∈ e.baseSet

Tactic:
]

Goal state after:
case refine_3
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
X : Type u_6
inst✝ : TopologicalSpace X
f : TotalSpace F E → X
s : Set B
hs : IsOpen[inst✝³] s
hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ univ)
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
z : TotalSpace F E
hz : z ∈ TotalSpace.proj ⁻¹' s
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ z.proj ∈ e.baseSet
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₁.baseSet

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₁.baseSet
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₂.baseSet

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₂.baseSet
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
p : Z
h : proj p ∈ e₁.baseSet
⊢ (↑e₂ p).2 = (↑e₂ p).2

Tactic:
]

Goal state after:
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
b : B
h : b ∈ e.baseSet
x : F
⊢ (b, x).2 = x

Tactic:
]

Goal state after:
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ e₃ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ (↑e₃ (↑e₁.symm (b, x))).2 = (↑e₃ (↑e₁.symm (b, x))).2

pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ e₃ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ ↑e₁.symm (b, x) ∈ e₂.source

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ e₃ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ ↑e₁.symm (b, x) ∈ e₂.source
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ e₃ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₂.baseSet

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ e₃ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₂.baseSet
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: case refine_3
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₂.baseSet

Tactic:
]

Goal state after:
case refine_3
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e₁ e₂ : Trivialization F proj
b : B
h₁ : b ∈ e₁.baseSet
h₂ : b ∈ e₂.baseSet
x : F
⊢ b ∈ e₂.baseSet
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Sort u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Set.range (toPullbackDiag f)) ↔ IsClosedEmbedding (toPullbackDiag f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Sort u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Set.range (toPullbackDiag f)) ↔ IsClosedEmbedding (toPullbackDiag f)
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal snd)

Tactic:
]

Goal state after:
X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal snd)
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.PullbackSelf.map_fst ⁻¹' Function.pullbackDiagonal f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.PullbackSelf.map_fst ⁻¹' Function.pullbackDiagonal f)
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
inj : Function.Injective g
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal (f ∘ g))

Tactic:
]

Goal state after:
X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
inj : Function.Injective g
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal (f ∘ g))
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
inj : Function.Injective g
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.mapPullback g id g ⋯ ⋯ ⁻¹' Function.pullbackDiagonal f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
inj : Function.Injective g
⊢ IsClosed[instTopologicalSpaceSubtype] (Function.mapPullback g id g ⋯ ⋯ ⁻¹' Function.pullbackDiagonal f)
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: case mp
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : IsLocallyInjective f
x : X
⊢ ∃ U ∈ 𝓝 x, Set.InjOn f U

pp: case mpr
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U
x : X
⊢ ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ Set.InjOn f U

Tactic:
· obtain ⟨U, ho, hm, hi⟩ := h x; exact ⟨U, ho.mem_nhds hm, hi⟩

Goal state after:
case mpr
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U
x : X
⊢ ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ Set.InjOn f U
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
n : ℕ
x y : A
⊢ ↑n x * y = x * ↑n y

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
n : ℕ
x y : A
⊢ ↑n x * y = x * ↑n y
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
n : ℤ
x y : A
⊢ ↑n x * y = x * ↑n y

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
n : ℤ
x y : A
⊢ ↑n x * y = x * ↑n y
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
s : Set B
⊢ proj ⁻¹' e.baseSet ∩ proj ⁻¹' frontier s = proj ⁻¹' e.baseSet ∩ proj ⁻¹' frontier s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsOpen[instTopologicalSpaceSubtype] (Set.range (toPullbackDiag f)) ↔ IsOpenEmbedding (toPullbackDiag f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsOpen[instTopologicalSpaceSubtype] (Set.range (toPullbackDiag f)) ↔ IsOpenEmbedding (toPullbackDiag f)
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd.1 ^ k) y
x y : A
⊢ (a.toProd.2 ^ (k + 1)) x * y = x * (a.toProd.1 ^ (k + 1)) y

Tactic:
]

Goal state after:
case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd.1 ^ k) y
x y : A
⊢ (a.toProd.2 ^ (k + 1)) x * y = x * (a.toProd.1 ^ (k + 1)) y
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
y : Y
⊢ (∀ (x : X), {x} ∈ 𝓝 x) ↔ ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn (fun x ↦ y) U

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
y : Y
⊢ (∀ (x : X), {x} ∈ 𝓝 x) ↔ ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn (fun x ↦ y) U
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
s : Set B
Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s
Heq : EqOn (↑e) (↑e') (proj ⁻¹' (e.baseSet ∩ frontier s))
⊢ proj ⁻¹' (e'.baseSet ∩ frontier s) = proj ⁻¹' (e'.baseSet ∩ frontier s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
s : Set B
Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s
Heq : EqOn (↑e) (↑e') (proj ⁻¹' (e.baseSet ∩ frontier s))
⊢ EqOn (↑e.toOpenPartialHomeomorph) (↑e'.toOpenPartialHomeomorph) (proj ⁻¹' (e.baseSet ∩ frontier s))

Tactic:
]

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
s : Set B
Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s
Heq : EqOn (↑e) (↑e') (proj ⁻¹' (e.baseSet ∩ frontier s))
⊢ EqOn (↑e.toOpenPartialHomeomorph) (↑e'.toOpenPartialHomeomorph) (proj ⁻¹' (e.baseSet ∩ frontier s))
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
hf : IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
hg : Function.Injective g
⊢ IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal (f ∘ g))

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
hf : IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
hg : Function.Injective g
⊢ IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal (f ∘ g))
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
hf : IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
hg : Function.Injective g
⊢ IsOpen[instTopologicalSpaceSubtype] (Function.mapPullback g id g ⋯ ⋯ ⁻¹' Function.pullbackDiagonal f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
hf : IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)
g : A → X
cont : Continuous[inst✝, inst✝¹] g
hg : Function.Injective g
⊢ IsOpen[instTopologicalSpaceSubtype] (Function.mapPullback g id g ⋯ ⋯ ⁻¹' Function.pullbackDiagonal f)
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd.1 ^ k) y
x y : A
⊢ x * (a.toProd.1 ^ k) (a.toProd.1 y) = x * (a.toProd.1 ^ k * a.toProd.1) y

Tactic:
mul_apply_eq_comp

Goal state after:
case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd.1 ^ k) y
x y : A
⊢ x * (a.toProd.1 ^ k) (a.toProd.1 y) = x * (a.toProd.1 ^ k) (a.toProd.1 y)
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd.1 ^ k) y
x y : A
⊢ x * (a.toProd.1 ^ k) (a.toProd.1 y) = x * (a.toProd.1 ^ k) (a.toProd.1 y)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : IsPreconnected s
h₁ : ContinuousOn g₁ s
h₂ : ContinuousOn g₂ s
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : IsPreconnected s
h₁ : ContinuousOn g₁ s
h₂ : ContinuousOn g₂ s
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : IsPreconnected s
h₁ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₁)
h₂ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₂)
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : IsPreconnected s
h₁ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₁)
h₂ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₂)
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : IsPreconnected s
h₁ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₁)
h₂ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₂)
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂

Tactic:
isPreconnected_iff_preconnectedSpace

Goal state after:
X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : PreconnectedSpace ↑s
h₁ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₁)
h₂ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₂)
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂