mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ ContinuousOn re ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' spectrum ℂ a)
ha : IsStarNormal a
⊢ ContinuousOn re ((fun x ↦ ↑x.re) '' spectrum ℂ a)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' spectrum ℂ a)
ha : IsStarNormal a
⊢ ContinuousOn re ((fun x ↦ ↑x.im) '' spectrum ℂ a)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (spectrum ℝ ↑(ℑ a))
ha : IsStarNormal a
this : ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' spectrum ℂ a)
| cfc f ↑(ℑ a)

Tactic:
rw [cfc_real_eq_complex, ← cfc_im_id a, ← cfc_comp' ..]
    simp

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (spectrum ℝ ↑(ℑ a))
ha : IsStarNormal a
this : ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' spectrum ℂ a)
| cfc (fun x ↦ ↑(f x.im)) a
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : Completion E
⊢ Continuous fun a ↦ ‖a‖ ^ 2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : Completion E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : Completion E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y z : Completion E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : Completion E
c : 𝕜
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
fun_prop

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

Goal state before:
pp: α : Type u_1
β : Type u_2
R₁ : Type u_3
R₂ : Type u_4
inst✝¹³ : UniformSpace α
inst✝¹² : AddCommGroup α
inst✝¹¹ : IsUniformAddGroup α
inst✝¹⁰ : Semiring R₁
inst✝⁹ : Module R₁ α
inst✝⁸ : UniformContinuousConstSMul R₁ α
inst✝⁷ : Semiring R₂
inst✝⁶ : UniformSpace β
inst✝⁵ : AddCommGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : Module R₂ β
inst✝² : UniformContinuousConstSMul R₂ β
σ : R₁ →+* R₂
inst✝¹ : T0Space β
inst✝ : CompleteSpace β
f : α →SL[σ] β
c : R₁
a : Completion α
⊢ Continuous fun a ↦ σ c • ((↑f).toAddMonoidHom.extension ⋯) a

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
⊢ ∀ (x y : f.PreGNS) (r : ℂ),
    f (star (f.ofPreGNS (r • x)) * f.ofPreGNS y) = (starRingEnd ℂ) r * f (star (f.ofPreGNS x) * f.ofPreGNS y)

Tactic:
by simp [smul_mul_assoc]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hf
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
x✝¹ x✝ : A
⊢ Continuous ⇑(f.leftMulMapPreGNS (x✝¹ + x✝)).completion

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
x✝¹ x✝ : A
⊢ Continuous ⇑((f.leftMulMapPreGNS x✝¹).completion + (f.leftMulMapPreGNS x✝).completion)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hf
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
x✝¹ x✝ : A
⊢ Continuous ⇑(f.leftMulMapPreGNS (x✝¹ * x✝)).completion

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
x✝¹ x✝ : A
⊢ Continuous ⇑((f.leftMulMapPreGNS x✝¹).completion * (f.leftMulMapPreGNS x✝).completion)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hf
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : A
⊢ Continuous fun y ↦ ⟪(f.leftMulMapPreGNS (star a)).completion y.1, y.2⟫_ℂ

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : A
⊢ Continuous fun y ↦ ⟪y.1, (f.leftMulMapPreGNS a).completion y.2⟫_ℂ

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hf
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A →ₚ[ℂ] ℂ
⊢ Continuous ⇑(f.gnsNonUnitalStarAlgHom 1)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp.hg
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A →ₚ[ℂ] ℂ
⊢ Continuous ⇑1

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Normed.Field.UnitBall

Goal state before:
pp: 𝕜✝ : Type u_1
𝕜 : Type u_2
inst✝² : SeminormedRing 𝕜
inst✝¹ : NormMulClass 𝕜
inst✝ : NormOneClass 𝕜
a✝ b✝ : 𝕜
hx : a✝ ∈ sphere 0 1
hy : b✝ ∈ sphere 0 1
⊢ a✝ * b✝ ∈ sphere 0 1

Tactic:
rw [mem_sphere_zero_iff_norm] at *

Goal state after:
𝕜✝ : Type u_1
𝕜 : Type u_2
inst✝² : SeminormedRing 𝕜
inst✝¹ : NormMulClass 𝕜
inst✝ : NormOneClass 𝕜
a✝ b✝ : 𝕜
hx : ‖a✝‖ = 1
hy : ‖b✝‖ = 1
⊢ ‖a✝ * b✝‖ = 1
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: x y : Circle
⊢ Continuous fun t ↦ cexp (↑t * I)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
x : X'
x₀ : X
hx₀ : x₀ ∈ e.source
hxx₀ : f x₀ = x
⊢ ↑e x₀ ∈ e.target

Tactic:
exact e.map_source' hx₀

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
x : X
⊢ extend f (↑e) (fun x ↦ Classical.arbitrary Z) (f x) = ↑e x

Tactic:
apply hf.injective.extend_apply

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✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
e✝ : Pretrivialization F proj
x : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] (Prod.map Subtype.val id)

Tactic:
fun_prop

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
v : F
hb : b ∈ e.baseSet
⊢ ContinuousAt (fun x ↦ (x, v)) b

Tactic:
fun_prop

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

Goal state before:
pp: B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
b : B
h : IsInducing (↑(a.pretrivializationAt b) ∘ TotalSpace.mk b)
⊢ IsInducing (TotalSpace.mk b)

Tactic:
by
  let := a.totalSpaceTopology
  rw [← domRestrict_comp_codRestrict (a.mem_pretrivializationAt_source b)] at h
  apply IsInducing.of_codRestrict (a.mem_pretrivializationAt_source b)
  refine h.of_comp ?_ (continuousOn_iff_continuous_domRestrict.mp
    (a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b)).continuousOn)
  exact (a.continuous_totalSpaceMk b).codRestrict (a.mem_pretrivializationAt_source b)

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

Goal state before:
pp: case refine_1
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
⊢ Continuous[inst✝¹, instTopologicalSpaceProd] (Prod.mk b)

Tactic:
fun_prop

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✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
b : B
y : E b
e : Trivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
⊢ Continuous[instTopologicalSpaceSubtype, inst✝²] Subtype.val

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u
A : Type v
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 b : 𝓜(𝕜, A)
h : a.toProd = b.toProd
⊢ a = b

Tactic:
by
  cases a
  cases b
  simpa using h

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))
⊢ e.source ∩ frontier (proj ⁻¹' s) = e'.source ∩ frontier (proj ⁻¹' s)

Tactic:
e.frontier_preimage,

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))
⊢ proj ⁻¹' (e.baseSet ∩ frontier s) = e'.source ∩ frontier (proj ⁻¹' 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:
← hg.preimage_pullbackDiagonal

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.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:
apply hf.preimage (cont.mapPullback cont)

Goal state after:
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
g₁ g₂ : A → X
h₁ : Continuous[inst✝, inst✝¹] g₁
h₂ : Continuous[inst✝, inst✝¹] g₂
sep : IsSeparatedMap f
he : f ∘ g₁ = f ∘ g₂
g : A → Function.Pullback f f := fun a ↦ ⟨(g₁ a, g₂ a), ⋯⟩
⊢ Continuous[inst✝, instTopologicalSpaceSubtype] g

Tactic:
fun_prop

Goal state after:
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
g₁ g₂ : A → X
h₁ : Continuous[inst✝, inst✝¹] g₁
h₂ : Continuous[inst✝, inst✝¹] g₂
inj : IsLocallyInjective f
he : f ∘ g₁ = f ∘ g₂
g : A → Function.Pullback f f := fun a ↦ ⟨(g₁ a, g₂ a), ⋯⟩
⊢ Continuous[inst✝, instTopologicalSpaceSubtype] g

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
h : IsLocalHomeomorphOn f s
hs : IsOpen[inst✝¹] s
⊢ DiscreteTopology ↑(f '' s) ↔ DiscreteTopology ↑s

Tactic:
refine ⟨fun _ ↦ h.discreteTopology_of_image, ?_⟩

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
h : IsLocalHomeomorphOn f s
hs : IsOpen[inst✝¹] s
⊢ DiscreteTopology ↑s → DiscreteTopology ↑(f '' s)
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
⊢ ↑𝕜 = ⇑(algebraMap 𝕜 𝓜(𝕜, 𝕜))

Tactic:
ext x : 3

Goal state after:
case fst
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
⊢ (↑𝕜 x).toProd.1 = ((algebraMap 𝕜 𝓜(𝕜, 𝕜)) x).toProd.1

case snd
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
⊢ (↑𝕜 x).toProd.2 = ((algebraMap 𝕜 𝓜(𝕜, 𝕜)) x).toProd.2
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
x y : A
⊢ Continuous fun x_1 ↦ (unop x_1.2) x * y

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
x y : A
⊢ Continuous fun x_1 ↦ x * x_1.1 y

Tactic:
fun_prop

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✝
T✝ : Trivialization F proj
z : Z
b : B
T : Trivialization F proj
⊢ Continuous[instTopologicalSpaceProd, inst✝³] fun x ↦ ↑x.2

Tactic:
fun_prop

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✝
T✝ : Trivialization F proj
z : Z
b : B
T : Trivialization F proj
⊢ Continuous[instTopologicalSpaceProd, inst✝¹] fun x ↦ ↑x.1

Tactic:
fun_prop

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✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
proj : Z → B
inst✝⁴ : TopologicalSpace Z
inst✝³ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
b✝ : B
y : E b✝
T✝ : Trivialization F proj
z : Z
b : B
ι : Type u_5
inst✝² : TopologicalSpace ι
inst✝¹ : LocallyCompactPair ι ↑T✝.baseSet
γ : C(ι, ↑T✝.baseSet)
i : ι
e : ↑T✝.source
T : Trivialization F proj
inst✝ : LocallyCompactPair ι ↑T.baseSet
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun eγt ↦ (eγt.1.2, eγt.2)

Tactic:
fun_prop

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✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
proj : Z → B
inst✝⁴ : TopologicalSpace Z
inst✝³ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
b✝ : B
y : E b✝
T✝ : Trivialization F proj
z : Z
b : B
ι : Type u_5
inst✝² : TopologicalSpace ι
inst✝¹ : LocallyCompactPair ι ↑T✝.baseSet
γ : C(ι, ↑T✝.baseSet)
i : ι
e : ↑T✝.source
T : Trivialization F proj
inst✝ : LocallyCompactPair ι ↑T.baseSet
Ψ : C((↑T.source × C(ι, ↑T.baseSet)) × ι, C(ι, ↑T.baseSet) × ι) :=
  { toFun := fun eγt ↦ (eγt.1.2, eγt.2), continuous_toFun := ⋯ }
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceSubtype] fun x ↦ x.1.1

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝ : TopologicalSpace I
x : X
h✝ : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[inst✝²] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ Continuous[inst✝, instTopologicalSpaceSubtype] fun i ↦ ⟨↑(H.symm (⟨x, hxU⟩, i)), ⋯⟩

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝ : TopologicalSpace I
x : X
h✝ : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[inst✝²] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ Continuous[instTopologicalSpaceSubtype, inst✝] fun e ↦ (H ⟨↑e, ⋯⟩).2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
inst✝ : Nonempty I
h✝ : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝²] U
hfU : IsOpen[inst✝³] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ Continuous[instTopologicalSpaceSubtype, instTopologicalSpaceProd] fun ui ↦ (⟨(↑ui).1, ⋯⟩, (↑ui).2)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝ : TopologicalSpace I
x : X
hxs : x ∈ s
h : IsEvenlyCovered f x I
inst : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[inst✝²] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ Continuous[instTopologicalSpaceSubtype, inst✝¹] Subtype.val

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
f : E → X
s : Set X
hf : IsCoveringMapOn f s
x : E
hx : x ∈ f ⁻¹' s
this : Nonempty ↑(f ⁻¹' {f x})
e : Trivialization (↑(f ⁻¹' {f x})) f := ⋯.toTrivialization
h : f x ∈ ⋯.toTrivialization.baseSet
he : x ∈ e.source := (Trivialization.mem_source e).mpr h
⊢ ContinuousOn (fun p ↦ (p, ⟨x, ⋯⟩)) e.baseSet

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
h :
  IsCoveringMap f ∧
    Function.Surjective f ∧
      ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
e : E
left✝ : DiscreteTopology ↑(f ⁻¹' {f e})
U : Set X
heU : f e ∈ U
hU : IsOpen[inst✝²] U
hfU : IsOpen[inst✝³] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × ↑(f ⁻¹' {f e})
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ Continuous[instTopologicalSpaceSubtype, instTopologicalSpaceSubtype] (Prod.snd ∘ ⇑H)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic

Goal state before:
pp: A : Type u_1
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
n : ℕ
a : A
ha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0
ha₁ : IsSelfAdjoint a
⊢ ContinuousOn Real.log (spectrum ℝ a)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsStrictlyPositive a
ha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0
⊢ ContinuousOn Real.exp (Real.log '' spectrum ℝ a)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: case refine_1.inr
A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ ≤ π
a✝ : Nontrivial A
h : -‖↑x‖ ∈ spectrum ℝ ↑x
⊢ Real.cos ‖x‖ ∈ (fun a ↦ Real.cos a) '' spectrum ℝ ↑x

Tactic:
| inr h =>

Goal state after: