Theorem: _private.Mathlib.CategoryTheory.Galois.Decomposition.0.CategoryTheory.PreGaloisCategory.selfProdTermIncl_fib_eq
File: Mathlib/CategoryTheory/Galois/Decomposition.lean

Proof steps:
state_before: C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
⊢ F.map u b = F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a
state_after: case h
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
⊢ ∀ (i : ↑(F.obj X)),
    (F.map (Pi.π (fun x => X) i)) (F.map u b) =
      (F.map (Pi.π (fun x => X) i)) (F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a)
tactic: apply Concrete.Pi.map_ext _ F
state_before: case h
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
⊢ ∀ (i : ↑(F.obj X)),
    (F.map (Pi.π (fun x => X) i)) (F.map u b) =
      (F.map (Pi.π (fun x => X) i)) (F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a)
state_after: case h
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
t : ↑(F.obj X)
⊢ (F.map (Pi.π (fun x => X) t)) (F.map u b) =
    (F.map (Pi.π (fun x => X) t)) (F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a)
tactic: intro (t : F.obj X)
state_before: case h
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
t : ↑(F.obj X)
⊢ (F.map (Pi.π (fun x => X) t)) (F.map u b) =
    (F.map (Pi.π (fun x => X) t)) (F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a)
state_after: case h.e'_2.h
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
t : ↑(F.obj X)
e_1✝ : (forget FintypeCat).obj (F.obj X) = ↑(F.obj X)
⊢ (F.map (Pi.π (fun x => X) t)) (F.map u b) = F.map (CategoryTheory.PreGaloisCategory.selfProdProj u t) b

case h.convert_2
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
t : ↑(F.obj X)
⊢ F.map (CategoryTheory.PreGaloisCategory.selfProdProj u t) b =
    (F.map (Pi.π (fun x => X) t)) (F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a)
tactic: convert_to F.map (selfProdProj u t) b = _