l3lab/ntp-mathlib huggingface dataset:
theorem isBounded_iff_subset_smul_ball {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.ball (0 : E) 1 := by
  rw [← isVonNBounded_iff 𝕜]
  constructor
  · intro h
    rcases h (Metric.ball_mem_nhds 0 zero_lt_one) with ⟨ρ, _, hρball⟩
    rcases NormedField.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩
    exact ⟨a, hρball a ha.le⟩
  · rintro ⟨a, ha⟩
    

Proof state:
case mpr.intro
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
a : 𝕜
ha : s ⊆ a • Metric.ball 0 1
⊢ Bornology.IsVonNBounded 𝕜 s

Next tactic:
exact ((isVonNBounded_ball 𝕜 E 1).image (a • (1 : E →L[𝕜] E))).subset ha
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by
  

Proof state:
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1

Next tactic:
constructor
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by
  constructor
  · 

Proof state:
case mp
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ Bornology.IsBounded s → ∃ a, s ⊆ a • Metric.closedBall 0 1

Next tactic:
rw [isBounded_iff_subset_smul_ball 𝕜]
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by
  constructor
  · rw [isBounded_iff_subset_smul_ball 𝕜]
    

Proof state:
case mp
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ (∃ a, s ⊆ a • Metric.ball 0 1) → ∃ a, s ⊆ a • Metric.closedBall 0 1

Next tactic:
exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by
  constructor
  · rw [isBounded_iff_subset_smul_ball 𝕜]
    exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall
  · 

Proof state:
case mpr
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsBounded s

Next tactic:
rw [← isVonNBounded_iff 𝕜]
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by
  constructor
  · rw [isBounded_iff_subset_smul_ball 𝕜]
    exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall
  · rw [← isVonNBounded_iff 𝕜]
    

Proof state:
case mpr
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsVonNBounded 𝕜 s

Next tactic:
rintro ⟨a, ha⟩
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by
  constructor
  · rw [isBounded_iff_subset_smul_ball 𝕜]
    exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall
  · rw [← isVonNBounded_iff 𝕜]
    rintro ⟨a, ha⟩
    

Proof state:
case mpr.intro
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
a : 𝕜
ha : s ⊆ a • Metric.closedBall 0 1
⊢ Bornology.IsVonNBounded 𝕜 s

Next tactic:
exact ((isVonNBounded_closedBall 𝕜 E 1).image (a • (1 : E →L[𝕜] E))).subset ha
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.)
-/
def toSheafify : F ⟶ F.sheafify.1 where
  app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
  naturality U U' f := by
    

Proof state:
X : TopCat
F : Presheaf (Type v) X
U U' : (Opens ↑X)ᵒᵖ
f : U ⟶ U'
⊢ F.map f ≫
      (fun U f =>
          { val := fun x => germ F x f,
            property :=
              (_ :
                PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate fun x =>
                  germ F x f) })
        U' =
    (fun U f =>
          { val := fun x => germ F x f,
            property :=
              (_ :
                PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate fun x =>
                  germ F x f) })
        U ≫
      (sheafify F).val.map f

Next tactic:
ext x
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.)
-/
def toSheafify : F ⟶ F.sheafify.1 where
  app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
  naturality U U' f := by
    ext x
    

Proof state:
case h
X : TopCat
F : Presheaf (Type v) X
U U' : (Opens ↑X)ᵒᵖ
f : U ⟶ U'
x : F.obj U
⊢ (F.map f ≫
        (fun U f =>
            { val := fun x => germ F x f,
              property :=
                (_ :
                  PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate fun x =>
                    germ F x f) })
          U')
      x =
    ((fun U f =>
            { val := fun x => germ F x f,
              property :=
                (_ :
                  PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate fun x =>
                    germ F x f) })
          U ≫
        (sheafify F).val.map f)
      x

Next tactic:
apply Subtype.ext
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.)
-/
def toSheafify : F ⟶ F.sheafify.1 where
  app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
  naturality U U' f := by
    ext x
    apply Subtype.ext -- Porting note: Added `apply`
    

Proof state:
case h.a
X : TopCat
F : Presheaf (Type v) X
U U' : (Opens ↑X)ᵒᵖ
f : U ⟶ U'
x : F.obj U
⊢ ↑((F.map f ≫
          (fun U f =>
              { val := fun x => germ F x f,
                property :=
                  (_ :
                    PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate fun x =>
                      germ F x f) })
            U')
        x) =
    ↑(((fun U f =>
              { val := fun x => germ F x f,
                property :=
                  (_ :
                    PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate fun x =>
                      germ F x f) })
            U ≫
          (sheafify F).val.map f)
        x)

Next tactic:
ext ⟨u, m⟩
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.)
-/
def toSheafify : F ⟶ F.sheafify.1 where
  app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
  naturality U U' f := by
    ext x
    apply Subtype.ext -- Porting note: Added `apply`
    ext ⟨u, m⟩
    

Proof state:
case h.a.h.mk
X : TopCat
F : Presheaf (Type v) X
U U' : (Opens ↑X)ᵒᵖ
f : U ⟶ U'
x : F.obj U
u : ↑X
m : u ∈ U'.unop
⊢ ↑((F.map f ≫
            (fun U f =>
                { val := fun x => germ F x f,
                  property :=
                    (_ :
                      PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate
                        fun x => germ F x f) })
              U')
          x)
      { val := u, property := m } =
    ↑(((fun U f =>
                { val := fun x => germ F x f,
                  property :=
                    (_ :
                      PrelocalPredicate.pred (PrelocalPredicate.sheafify (Sheafify.isGerm F)).toPrelocalPredicate
                        fun x => germ F x f) })
              U ≫
            (sheafify F).val.map f)
          x)
      { val := u, property := m }

Next tactic:
exact germ_res_apply F f.unop ⟨u, m⟩ x
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  

Proof state:
X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ Function.Surjective (stalkToFiber F x)

Next tactic:
apply TopCat.stalkToFiber_surjective
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_surjective
  

Proof state:
case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ ∀ (t : stalk F x),
    ∃ U f,
      ∃ (_ : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate f),
        f { val := x, property := (_ : x ∈ U.obj) } = t

Next tactic:
intro t
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_surjective
  intro t
  

Proof state:
case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
t : stalk F x
⊢ ∃ U f,
    ∃ (_ : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate f),
      f { val := x, property := (_ : x ∈ U.obj) } = t

Next tactic:
obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_surjective
  intro t
  obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t
  · 

Proof state:
case w.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : (forget (Type v)).obj (F.obj (op U))
⊢ ∃ U_1 f,
    ∃ (_ : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate f),
      f { val := x, property := (_ : x ∈ U_1.obj) } = (germ F { val := x, property := m }) s

Next tactic:
use ⟨U, m⟩
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_surjective
  intro t
  obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t
  · use ⟨U, m⟩
    

Proof state:
case h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : (forget (Type v)).obj (F.obj (op U))
⊢ ∃ f,
    ∃ (_ : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate f),
      f { val := x, property := (_ : x ∈ { obj := U, property := m }.obj) } = (germ F { val := x, property := m }) s

Next tactic:
fconstructor
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_surjective
  intro t
  obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t
  · use ⟨U, m⟩
    fconstructor
    · 

Proof state:
case h.w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : (forget (Type v)).obj (F.obj (op U))
⊢ (y : ↥{ obj := U, property := m }.obj) → stalk F ↑y

Next tactic:
exact fun y => F.germ y s
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_surjective
  intro t
  obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t
  · use ⟨U, m⟩
    fconstructor
    · exact fun y => F.germ y s
    · 

Proof state:
case h.h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : (forget (Type v)).obj (F.obj (op U))
⊢ ∃ (_ : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fun y => germ F y s),
    germ F { val := x, property := (_ : x ∈ { obj := U, property := m }.obj) } s =
      (germ F { val := x, property := m }) s

Next tactic:
exact ⟨PrelocalPredicate.sheafifyOf ⟨s, fun _ => rfl⟩, rfl⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  

Proof state:
X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ Function.Injective (stalkToFiber F x)

Next tactic:
apply TopCat.stalkToFiber_injective
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  

Proof state:
case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → stalk F ↑y),
    PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU →
      ∀ (fV : (y : ↥V.obj) → stalk F ↑y),
        PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV →
          fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } →
            ∃ W iU iV,
              ∀ (w : ↥W.obj),
                fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) =
                  fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w)

Next tactic:
intro U V fU hU fV hV e
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  

Proof state:
case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
e : fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) }
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  

Proof state:
case w.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
e : fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) }
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
e : fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) }
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
have wUx := wU ⟨x, mU⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
e : fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) }
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
wUx :
  (fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.obj) }) x_1)) { val := x, property := mU } =
    germ F { val := x, property := mU } gU
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
dsimp at wUx
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; 

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
e : fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) }
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
wUx : fU { val := x, property := (_ : ↑{ val := x, property := mU } ∈ ↑U.obj) } = germ F { val := x, property := mU } gU
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
erw [wUx] at e
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; 

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
e : germ F { val := x, property := mU } gU = fV { val := x, property := (_ : x ∈ V.obj) }
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
wUx : fU { val := x, property := (_ : ↑{ val := x, property := mU } ∈ ↑U.obj) } = germ F { val := x, property := mU } gU
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
clear wUx
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
e : germ F { val := x, property := mU } gU = fV { val := x, property := (_ : x ∈ V.obj) }
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
have wVx := wV ⟨x, mV⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
e : germ F { val := x, property := mU } gU = fV { val := x, property := (_ : x ∈ V.obj) }
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
wVx :
  (fun x_1 => fV ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑V.obj) }) x_1)) { val := x, property := mV } =
    germ F { val := x, property := mV } gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
dsimp at wVx