ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StructureGroupoid.LocalInvariantProp.liftPropAt_chart_symm
theorem liftPropAt_chart_symm [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
    (hQ : ∀ y, Q id univ y) : LiftPropAt Q (chartAt (H := H) x).symm ((chartAt H x) x) 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {x : M} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y)
Goal: ChartedSpace.LiftPropAt Q (↑(chartAt H x).symm) (↑(chartAt H x) x)
-- StructureGroupoid.LocalInvariantProp.liftPropOn_chart_symm
theorem liftPropOn_chart_symm [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
    (hQ : ∀ y, Q id univ y) : LiftPropOn Q (chartAt (H := H) x).symm (chartAt H x).target 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {x : M} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y)
Goal: ChartedSpace.LiftPropOn Q (↑(chartAt H x).symm) (chartAt H x).target
-- StructureGroupoid.LocalInvariantProp.liftPropAt_of_mem_groupoid
theorem liftPropAt_of_mem_groupoid (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y)
    {f : PartialHomeomorph H H} (hf : f ∈ G) {x : H} (hx : x ∈ f.source) : LiftPropAt Q f x 
Arguments: {H : Type u_1} [TopologicalSpace H] {G : StructureGroupoid H} {Q : (H → H) → Set H → H → Prop} (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) {f : PartialHomeomorph H H} (hf : f ∈ G) {x : H} (hx : x ∈ f.source)
Goal: ChartedSpace.LiftPropAt Q (↑f) x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_groupoid
theorem liftPropOn_of_mem_groupoid (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y)
    {f : PartialHomeomorph H H} (hf : f ∈ G) : LiftPropOn Q f f.source 
Arguments: {H : Type u_1} [TopologicalSpace H] {G : StructureGroupoid H} {Q : (H → H) → Set H → H → Prop} (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) {f : PartialHomeomorph H H} (hf : f ∈ G)
Goal: ChartedSpace.LiftPropOn Q (↑f) f.source
-- StructureGroupoid.LocalInvariantProp.liftProp_id
theorem liftProp_id (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) :
    LiftProp Q (id : M → M) 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {Q : (H → H) → Set H → H → Prop} (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y)
Goal: ChartedSpace.LiftProp Q id
-- StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_subtype_val
theorem liftPropAt_iff_comp_subtype_val (hG : LocalInvariantProp G G' P) {U : Opens M}
    (f : M → M') (x : U) :
    LiftPropAt P f x ↔ LiftPropAt P (f ∘ Subtype.val) x 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {U : TopologicalSpace.Opens M} (f : M → M') (x : ↥U)
Goal: ChartedSpace.LiftPropAt P f ↑x ↔ ChartedSpace.LiftPropAt P (f ∘ Subtype.val) x
-- StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_inclusion
theorem liftPropAt_iff_comp_inclusion (hG : LocalInvariantProp G G' P) {U V : Opens M} (hUV : U ≤ V)
    (f : V → M') (x : U) :
    LiftPropAt P f (Set.inclusion hUV x) ↔ LiftPropAt P (f ∘ Set.inclusion hUV : U → M') x 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {U V : TopologicalSpace.Opens M} (hUV : U ≤ V) (f : ↥V → M') (x : ↥U)
Goal: ChartedSpace.LiftPropAt P f (Set.inclusion hUV x) ↔ ChartedSpace.LiftPropAt P (f ∘ Set.inclusion hUV) x
-- StructureGroupoid.LocalInvariantProp.liftProp_subtype_val
theorem liftProp_subtype_val {Q : (H → H) → Set H → H → Prop} (hG : LocalInvariantProp G G Q)
    (hQ : ∀ y, Q id univ y) (U : Opens M) :
    LiftProp Q (Subtype.val : U → M) 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {Q : (H → H) → Set H → H → Prop} (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) (U : TopologicalSpace.Opens M)
Goal: ChartedSpace.LiftProp Q Subtype.val
-- StructureGroupoid.LocalInvariantProp.liftProp_inclusion
theorem liftProp_inclusion {Q : (H → H) → Set H → H → Prop} (hG : LocalInvariantProp G G Q)
    (hQ : ∀ y, Q id univ y) {U V : Opens M} (hUV : U ≤ V) :
    LiftProp Q (Set.inclusion hUV : U → V) 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {Q : (H → H) → Set H → H → Prop} (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) {U V : TopologicalSpace.Opens M} (hUV : U ≤ V)
Goal: ChartedSpace.LiftProp Q (Set.inclusion hUV)
-- StructureGroupoid.isLocalStructomorphWithinAt_localInvariantProp
theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
    LocalInvariantProp G G (IsLocalStructomorphWithinAt G) 
Arguments: {H : Type u_1} [TopologicalSpace H] (G : StructureGroupoid H) [ClosedUnderRestriction G]
Goal: G.LocalInvariantProp G G.IsLocalStructomorphWithinAt
-- PartialHomeomorph.isLocalStructomorphWithinAt_iff
theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff {G : StructureGroupoid H}
    [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H}
    (hx : x ∈ f.source ∪ sᶜ) :
    G.IsLocalStructomorphWithinAt (⇑f) s x ↔
      x ∈ s → ∃ e : PartialHomeomorph H H,
      e ∈ G ∧ e.source ⊆ f.source ∧ EqOn f (⇑e) (s ∩ e.source) ∧ x ∈ e.source 
Arguments: {H : Type u_1} [TopologicalSpace H] {G : StructureGroupoid H} [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H} (hx : x ∈ f.source ∪ sᶜ)
Goal: G.IsLocalStructomorphWithinAt (↑f) s x ↔
  x ∈ s → ∃ e ∈ G, e.source ⊆ f.source ∧ Set.EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source
-- PartialHomeomorph.isLocalStructomorphWithinAt_iff'
theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff' {G : StructureGroupoid H}
    [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H} (hs : f.source ⊆ s)
    (hx : x ∈ f.source ∪ sᶜ) :
    G.IsLocalStructomorphWithinAt (⇑f) s x ↔
      x ∈ s → ∃ e : PartialHomeomorph H H,
      e ∈ G ∧ e.source ⊆ f.source ∧ EqOn f (⇑e) e.source ∧ x ∈ e.source 
Arguments: {H : Type u_1} [TopologicalSpace H] {G : StructureGroupoid H} [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H} (hs : f.source ⊆ s) (hx : x ∈ f.source ∪ sᶜ)
Goal: G.IsLocalStructomorphWithinAt (↑f) s x ↔
  x ∈ s → ∃ e ∈ G, e.source ⊆ f.source ∧ Set.EqOn (↑f) (↑e) e.source ∧ x ∈ e.source
-- PartialHomeomorph.isLocalStructomorphWithinAt_source_iff
theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_source_iff {G : StructureGroupoid H}
    [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {x : H} :
    G.IsLocalStructomorphWithinAt (⇑f) f.source x ↔
      x ∈ f.source → ∃ e : PartialHomeomorph H H,
      e ∈ G ∧ e.source ⊆ f.source ∧ EqOn f (⇑e) e.source ∧ x ∈ e.source 
Arguments: {H : Type u_1} [TopologicalSpace H] {G : StructureGroupoid H} [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {x : H}
Goal: G.IsLocalStructomorphWithinAt (↑f) f.source x ↔
  x ∈ f.source → ∃ e ∈ G, e.source ⊆ f.source ∧ Set.EqOn (↑f) (↑e) e.source ∧ x ∈ e.source
-- StructureGroupoid.HasGroupoid.comp
theorem HasGroupoid.comp
    (H : ∀ e ∈ G₂, LiftPropOn (IsLocalStructomorphWithinAt G₁) (e : H₂ → H₂) e.source) :
    @HasGroupoid H₁ _ H₃ _ (ChartedSpace.comp H₁ H₂ H₃) G₁ 
Arguments: {H₁ : Type u_6} [TopologicalSpace H₁] {H₂ : Type u_7} [TopologicalSpace H₂] {H₃ : Type u_8} [TopologicalSpace H₃] [ChartedSpace H₁ H₂] [ChartedSpace H₂ H₃] {G₁ : StructureGroupoid H₁} [HasGroupoid H₂ G₁] [ClosedUnderRestriction G₁] (G₂ : StructureGroupoid H₂) [HasGroupoid H₃ G₂] (H : ∀ e ∈ G₂, ChartedSpace.LiftPropOn G₁.IsLocalStructomorphWithinAt (↑e) e.source)
Goal: HasGroupoid H₃ G₁
-- AlgebraicGeometry.SheafedSpace.IsSheaf
structure SheafedSpace extends PresheafedSpace C where
    IsSheaf : presheaf.IsSheaf
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (self : AlgebraicGeometry.SheafedSpace C)
Goal: self.presheaf.IsSheaf
-- AlgebraicGeometry.SheafedSpace.mk_coe
theorem mk_coe (carrier) (presheaf) (h) :
    (({ carrier
        presheaf
        IsSheaf := h } : SheafedSpace C) : TopCat) = carrier 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (carrier : TopCat) (presheaf : TopCat.Presheaf C carrier) (h : { carrier := carrier, presheaf := presheaf }.presheaf.IsSheaf)
Goal: ↑{ carrier := carrier, presheaf := presheaf, IsSheaf := h }.toPresheafedSpace = carrier
-- AlgebraicGeometry.SheafedSpace.ext
@[ext]
theorem ext {X Y : SheafedSpace C} {α β : X ⟶ Y} (w : α.base = β.base)
    (h : α.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.c) : α = β 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : AlgebraicGeometry.SheafedSpace C} {α β : X ⟶ Y} (w : α.base = β.base) (h : α.c ≫ CategoryTheory.whiskerRight (CategoryTheory.eqToHom ⋯) X.presheaf = β.c)
Goal: α = β
-- AlgebraicGeometry.SheafedSpace.isoMk_hom
@[simps]
def isoMk {X Y : SheafedSpace C} (e : X.toPresheafedSpace ≅ Y.toPresheafedSpace) : X ≅ Y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : AlgebraicGeometry.SheafedSpace C} (e : X.toPresheafedSpace ≅ Y.toPresheafedSpace)
Goal: (AlgebraicGeometry.SheafedSpace.isoMk e).hom = e.hom
-- AlgebraicGeometry.SheafedSpace.isoMk_inv
@[simps]
def isoMk {X Y : SheafedSpace C} (e : X.toPresheafedSpace ≅ Y.toPresheafedSpace) : X ≅ Y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : AlgebraicGeometry.SheafedSpace C} (e : X.toPresheafedSpace ≅ Y.toPresheafedSpace)
Goal: (AlgebraicGeometry.SheafedSpace.isoMk e).inv = e.inv
-- AlgebraicGeometry.SheafedSpace.forgetToPresheafedSpace_obj
@[simps! obj map]
def forgetToPresheafedSpace : SheafedSpace C ⥤ PresheafedSpace C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (self : AlgebraicGeometry.SheafedSpace C)
Goal: AlgebraicGeometry.SheafedSpace.forgetToPresheafedSpace.obj self = self.toPresheafedSpace
-- AlgebraicGeometry.SheafedSpace.forgetToPresheafedSpace_map
@[simps! obj map]
def forgetToPresheafedSpace : SheafedSpace C ⥤ PresheafedSpace C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C]
Goal: ∀
  {X Y :
    CategoryTheory.InducedCategory (AlgebraicGeometry.PresheafedSpace C)
      AlgebraicGeometry.SheafedSpace.toPresheafedSpace}
  (f : X ⟶ Y), AlgebraicGeometry.SheafedSpace.forgetToPresheafedSpace.map f = f
-- AlgebraicGeometry.SheafedSpace.id_base
@[simp]
theorem id_base (X : SheafedSpace C) : (𝟙 X : X ⟶ X).base = 𝟙 (X : TopCat) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.SheafedSpace C)
Goal: (𝟙 X).base = 𝟙 ↑X.toPresheafedSpace
-- AlgebraicGeometry.SheafedSpace.id_c
theorem id_c (X : SheafedSpace C) :
    (𝟙 X : X ⟶ X).c = eqToHom (Presheaf.Pushforward.id_eq X.presheaf).symm 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.SheafedSpace C)
Goal: (𝟙 X).c = CategoryTheory.eqToHom ⋯
-- AlgebraicGeometry.SheafedSpace.id_c_app
@[simp]
theorem id_c_app (X : SheafedSpace C) (U) :
    (𝟙 X : X ⟶ X).c.app U = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.SheafedSpace C) (U : (TopologicalSpace.Opens ↑↑X.toPresheafedSpace)ᵒᵖ)
Goal: (𝟙 X).c.app U = 𝟙 (X.presheaf.obj U)
-- AlgebraicGeometry.SheafedSpace.comp_base
@[simp]
theorem comp_base {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) :
    (f ≫ g).base = f.base ≫ g.base 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : AlgebraicGeometry.SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (f ≫ g).base = f.base ≫ g.base
-- AlgebraicGeometry.SheafedSpace.comp_c_app
@[simp]
theorem comp_c_app {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :
    (α ≫ β).c.app U = β.c.app U ≫ α.c.app (op ((Opens.map β.base).obj (unop U))) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : AlgebraicGeometry.SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U : (TopologicalSpace.Opens ↑↑Z.toPresheafedSpace)ᵒᵖ)
Goal: (α ≫ β).c.app U = β.c.app U ≫ α.c.app (Opposite.op ((TopologicalSpace.Opens.map β.base).obj (Opposite.unop U)))
-- AlgebraicGeometry.SheafedSpace.comp_c_app'
theorem comp_c_app' {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :
    (α ≫ β).c.app (op U) = β.c.app (op U) ≫ α.c.app (op ((Opens.map β.base).obj U)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : AlgebraicGeometry.SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U : TopologicalSpace.Opens ↑↑Z.toPresheafedSpace)
Goal: (α ≫ β).c.app (Opposite.op U) =
  β.c.app (Opposite.op U) ≫ α.c.app (Opposite.op ((TopologicalSpace.Opens.map β.base).obj U))
-- AlgebraicGeometry.SheafedSpace.congr_app
theorem congr_app {X Y : SheafedSpace C} {α β : X ⟶ Y} (h : α = β) (U) :
    α.c.app U = β.c.app U ≫ X.presheaf.map (eqToHom (by subst h; rfl)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : AlgebraicGeometry.SheafedSpace C} {α β : X ⟶ Y} (h : α = β) (U : (TopologicalSpace.Opens ↑↑Y.toPresheafedSpace)ᵒᵖ)
Goal: α.c.app U = β.c.app U ≫ X.presheaf.map (CategoryTheory.eqToHom ⋯)
-- AlgebraicGeometry.SheafedSpace.ofRestrict_c_app
@[simps!]
def ofRestrict {U : TopCat} (X : SheafedSpace C) {f : U ⟶ (X : TopCat)}
    (h : OpenEmbedding f) : X.restrict h ⟶ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {U : TopCat} (X : AlgebraicGeometry.SheafedSpace C) {f : U ⟶ ↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : (TopologicalSpace.Opens ↑↑X.toPresheafedSpace)ᵒᵖ)
Goal: (X.ofRestrict h).c.app V = X.presheaf.map (⋯.adjunction.counit.app (Opposite.unop V)).op
-- AlgebraicGeometry.SheafedSpace.ofRestrict_base
@[simps!]
def ofRestrict {U : TopCat} (X : SheafedSpace C) {f : U ⟶ (X : TopCat)}
    (h : OpenEmbedding f) : X.restrict h ⟶ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {U : TopCat} (X : AlgebraicGeometry.SheafedSpace C) {f : U ⟶ ↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f)
Goal: (X.ofRestrict h).base = f
-- AlgebraicGeometry.SheafedSpace.restrictTopIso_inv
@[simps! hom inv]
def restrictTopIso (X : SheafedSpace C) : X.restrict (Opens.openEmbedding ⊤) ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.SheafedSpace C)
Goal: X.restrictTopIso.inv = X.toRestrictTop
-- AlgebraicGeometry.SheafedSpace.restrictTopIso_hom
@[simps! hom inv]
def restrictTopIso (X : SheafedSpace C) : X.restrict (Opens.openEmbedding ⊤) ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.SheafedSpace C)
Goal: X.restrictTopIso.hom = X.ofRestrict ⋯
-- AlgebraicGeometry.SheafedSpace.Γ_def
theorem Γ_def : (Γ : _ ⥤ C) = forgetToPresheafedSpace.op ⋙ PresheafedSpace.Γ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C]
Goal: AlgebraicGeometry.SheafedSpace.Γ =
  AlgebraicGeometry.SheafedSpace.forgetToPresheafedSpace.op ⋙ AlgebraicGeometry.PresheafedSpace.Γ
-- AlgebraicGeometry.SheafedSpace.Γ_obj
@[simp]
theorem Γ_obj (X : (SheafedSpace C)ᵒᵖ) : Γ.obj X = (unop X).presheaf.obj (op ⊤) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : (AlgebraicGeometry.SheafedSpace C)ᵒᵖ)
Goal: AlgebraicGeometry.SheafedSpace.Γ.obj X = (Opposite.unop X).presheaf.obj (Opposite.op ⊤)
-- AlgebraicGeometry.SheafedSpace.Γ_obj_op
theorem Γ_obj_op (X : SheafedSpace C) : Γ.obj (op X) = X.presheaf.obj (op ⊤) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.SheafedSpace C)
Goal: AlgebraicGeometry.SheafedSpace.Γ.obj (Opposite.op X) = X.presheaf.obj (Opposite.op ⊤)
-- AlgebraicGeometry.SheafedSpace.Γ_map
@[simp]
theorem Γ_map {X Y : (SheafedSpace C)ᵒᵖ} (f : X ⟶ Y) : Γ.map f = f.unop.c.app (op ⊤) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : (AlgebraicGeometry.SheafedSpace C)ᵒᵖ} (f : X ⟶ Y)
Goal: AlgebraicGeometry.SheafedSpace.Γ.map f = f.unop.c.app (Opposite.op ⊤)
-- AlgebraicGeometry.SheafedSpace.Γ_map_op
theorem Γ_map_op {X Y : SheafedSpace C} (f : X ⟶ Y) : Γ.map f.op = f.c.app (op ⊤) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : AlgebraicGeometry.SheafedSpace C} (f : X ⟶ Y)
Goal: AlgebraicGeometry.SheafedSpace.Γ.map f.op = f.c.app (Opposite.op ⊤)
-- AlgebraicGeometry.SheafedSpace.hom_stalk_ext
lemma hom_stalk_ext {X Y : SheafedSpace C} (f g : X ⟶ Y) (h : f.base = g.base)
    (h' : ∀ x, f.stalkMap x = (Y.presheaf.stalkCongr (h ▸ rfl)).hom ≫ g.stalkMap x) :
    f = g 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasColimits C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] [(CategoryTheory.forget C).ReflectsIsomorphisms] {X Y : AlgebraicGeometry.SheafedSpace C} (f g : X ⟶ Y) (h : f.base = g.base) (h' :
  ∀ (x : ↑↑X.toPresheafedSpace),
    AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f x =
      (Y.presheaf.stalkCongr ⋯).hom ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap g x)
Goal: f = g
-- AlgebraicGeometry.SheafedSpace.mono_of_base_injective_of_stalk_epi
lemma mono_of_base_injective_of_stalk_epi {X Y : SheafedSpace C} (f : X ⟶ Y)
    (h₁ : Function.Injective f.base)
    (h₂ : ∀ x, Epi (f.stalkMap x)) : Mono f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasColimits C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] [(CategoryTheory.forget C).ReflectsIsomorphisms] {X Y : AlgebraicGeometry.SheafedSpace C} (f : X ⟶ Y) (h₁ : Function.Injective ⇑f.base) (h₂ : ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.Epi (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f x))
Goal: CategoryTheory.Mono f
-- FiberBundle.chartedSpace'_chartAt
theorem FiberBundle.chartedSpace'_chartAt (x : TotalSpace F E) :
    chartAt (B × F) x = (trivializationAt F E x.proj).toPartialHomeomorph 
Arguments: {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [TopologicalSpace B] [FiberBundle F E] (x : Bundle.TotalSpace F E)
Goal: chartAt (B × F) x = (trivializationAt F E x.proj).toPartialHomeomorph
-- FiberBundle.chartedSpace_chartAt
theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) :
    chartAt (ModelProd HB F) x =
      (trivializationAt F E x.proj).toPartialHomeomorph ≫ₕ
        (chartAt HB x.proj).prod (PartialHomeomorph.refl F) 
Arguments: {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {HB : Type u_7} [TopologicalSpace HB] [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (x : Bundle.TotalSpace F E)
Goal: chartAt (ModelProd HB F) x =
  (trivializationAt F E x.proj).toPartialHomeomorph ≫ₕ (chartAt HB x.proj).prod (PartialHomeomorph.refl F)
-- FiberBundle.chartedSpace_chartAt_symm_fst
theorem FiberBundle.chartedSpace_chartAt_symm_fst (x : TotalSpace F E) (y : ModelProd HB F)
    (hy : y ∈ (chartAt (ModelProd HB F) x).target) :
    ((chartAt (ModelProd HB F) x).symm y).proj = (chartAt HB x.proj).symm y.1 
Arguments: {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {HB : Type u_7} [TopologicalSpace HB] [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (x : Bundle.TotalSpace F E) (y : ModelProd HB F) (hy : y ∈ (chartAt (ModelProd HB F) x).target)
Goal: (↑(chartAt (ModelProd HB F) x).symm y).proj = ↑(chartAt HB x.proj).symm y.1
-- FiberBundle.extChartAt
protected theorem FiberBundle.extChartAt (x : TotalSpace F E) :
    extChartAt (IB.prod 𝓘(𝕜, F)) x =
      (trivializationAt F E x.proj).toPartialEquiv ≫
        (extChartAt IB x.proj).prod (PartialEquiv.refl F) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (x : Bundle.TotalSpace F E)
Goal: extChartAt (IB.prod 𝓘(𝕜, F)) x =
  (trivializationAt F E x.proj).toPartialEquiv ≫ (extChartAt IB x.proj).prod (PartialEquiv.refl F)
-- FiberBundle.extChartAt_target
protected theorem FiberBundle.extChartAt_target (x : TotalSpace F E) :
    (extChartAt (IB.prod 𝓘(𝕜, F)) x).target =
      ((extChartAt IB x.proj).target ∩
        (extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ univ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (x : Bundle.TotalSpace F E)
Goal: (extChartAt (IB.prod 𝓘(𝕜, F)) x).target =
  ((extChartAt IB x.proj).target ∩ ↑(extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ Set.univ
-- FiberBundle.writtenInExtChartAt_trivializationAt
theorem FiberBundle.writtenInExtChartAt_trivializationAt {x : TotalSpace F E} {y}
    (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target) :
    writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) x
      (trivializationAt F E x.proj) y = y 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {x : Bundle.TotalSpace F E} {y : EB × F} (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target)
Goal: writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) x (↑(trivializationAt F E x.proj)) y = y
-- FiberBundle.writtenInExtChartAt_trivializationAt_symm
theorem FiberBundle.writtenInExtChartAt_trivializationAt_symm {x : TotalSpace F E} {y}
    (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target) :
    writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (trivializationAt F E x.proj x)
      (trivializationAt F E x.proj).toPartialHomeomorph.symm y = y 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {x : Bundle.TotalSpace F E} {y : EB × F} (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target)
Goal: writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑(trivializationAt F E x.proj) x)
    (↑(trivializationAt F E x.proj).symm) y =
  y