-- Module: Mathlib.AlgebraicGeometry.Group.Abelian
Goal:
K : Type u
inst : Field K
inst_1 : IsAlgClosed K
G : Over (Spec (CommRingCat.of K))
inst_2 : IsProper G.hom
inst_3 : IsIntegral.{u} (tensorObj (𝒞 := instCategoryOver) G G).left
inst_4 : GrpObj G
S : Scheme := Spec (CommRingCat.of K)
point : ↥S := IsLocalRing.closedPoint K
hpoint : IsClosed (X := ↥S) {point}
this_1 : Nonempty.{u + 1} ↥G.left
this_2 : IsProper (tensorObj (𝒞 := instCategoryOver) G G).hom
this_3 : JacobsonSpace.{u} ↥(tensorObj (𝒞 := instCategoryOver) G G).left
this_4 : Surjective G.hom
this_5 : IsProper (X := (tensorObj (𝒞 := instCategoryOver) G G).left) (Over.Hom.left (fst G G))
this_6 : Surjective (X := (tensorObj (𝒞 := instCategoryOver) G G).left) (Over.Hom.left (fst G G))
this_7 : IsProper (X := (tensorObj (𝒞 := instCategoryOver) G G).left) (Over.Hom.left (GrpObj.commutator G) ≫ G.hom)
this_8 :
  IsClosedImmersion (X :=
    Over.left (X := Spec (CommRingCat.of K)) ((𝟙_ (Over (Spec (CommRingCat.of K)))) (𝒞 := instCategoryOver)))
    (Over.Hom.left (lift η η) ≫ Over.Hom.left (fst G G))
this_9 :
  IsClosedImmersion.{u} (X :=
    Over.left (X := Spec (CommRingCat.of K)) ((𝟙_ (Over (Spec (CommRingCat.of K)))) (𝒞 := instCategoryOver))) (Y :=
    (tensorObj (𝒞 := instCategoryOver) G G).left) (Over.Hom.left (lift η η))
γ : tensorObj (𝒞 := instCategoryOver) G G ⟶ tensorObj (𝒞 := instCategoryOver) G G :=
  lift (fst G G) (GrpObj.commutator G)
this_10 : IsProper (Over.Hom.left γ ≫ Over.Hom.left (fst G G))
this_11 : IsProper (Over.Hom.left γ)
H_1 :
  Subset (α := Set ↥(tensorObj (𝒞 := instCategoryOver) G G).left)
    (Set.image (α := ↥(tensorObj (𝒞 := instCategoryOver) G G).left) (⇑(Over.Hom.left γ))
      (Set.preimage (β := ↥G.left) (⇑(Over.Hom.left (fst G G)))
        (singleton (α := ↥G.left) ((Over.Hom.left η) point : ↥G.left))))
    (singleton (α := ↥(tensorObj (𝒞 := instCategoryOver) G G).left)
      ((Over.Hom.left (lift η η)) point : ↥(tensorObj (𝒞 := instCategoryOver) G G).left))
U : G.left.Opens
hηU : Membership.mem (α := ↥G.left) U ((Over.Hom.left η) point : ↥G.left)
H : IsFinite ((Scheme.Hom.imageι (Over.Hom.left γ) ≫ Over.Hom.left (fst G G)) ∣_ U)
x : ↥U
this : Eq (α := Set ↥(↑U : Scheme)) (Set.preimage (β := ↥G.left) ⇑U.ι {(↑x : ↥G.left)}) {x}
simp :
  Subset (α := Set ↥(tensorObj (𝒞 := instCategoryOver) G G).left)
    (Set.preimage (β := ↥G.left) ⇑(pullback.fst G.hom G.hom) {(↑x : ↥G.left)} ∩
      Set.range («ι» := ↥(tensorObj (𝒞 := instCategoryOver) G G).left) ⇑(Over.Hom.left γ))
    (Set.preimage (β := ↥G.left) ⇑(Over.Hom.left (fst G G)) {(↑x : ↥G.left)} ∩
      ⇑(Scheme.Hom.imageι (Over.Hom.left γ)) ''
        (↑((TopologicalSpace.Opens.map
                  ((Scheme.Hom.imageι (Over.Hom.left γ)).base ≫
                    PresheafedSpace.Hom.base
                      (LocallyRingedSpace.Hom.toHom (Scheme.Hom.toLRSHom' (Over.Hom.left (fst G G)))))).obj
              U) :
          Set ↥(Scheme.Hom.image (Over.Hom.left γ))))
⊢ Subset (α := Set ↥(tensorObj (𝒞 := instCategoryOver) G G).left)
    (Set.preimage (β := ↥G.left) ⇑(pullback.fst G.hom G.hom) {(↑x : ↥G.left)} ∩
      Set.range («ι» := ↥(tensorObj (𝒞 := instCategoryOver) G G).left) ⇑(Over.Hom.left γ))
    (Set.preimage (β := ↥G.left) ⇑(Over.Hom.left (fst G G)) {(↑x : ↥G.left)} ∩
      Set.range («ι» := ↥(↑((Scheme.Hom.imageι (Over.Hom.left γ) ≫ Over.Hom.left (fst G G)) ⁻¹ᵁ U) : Scheme))
        ⇑(((Scheme.Hom.imageι (Over.Hom.left γ) ≫ Over.Hom.left (fst G G)) ⁻¹ᵁ U).ι ≫
            Scheme.Hom.imageι (Over.Hom.left γ)))

Tactic:
simp only [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp, Scheme.Opens.range_ι]

Premises used: Set.range_comp, AlgebraicGeometry.Scheme.Opens.range_ι, AlgebraicGeometry.Scheme.Hom.comp_base, TopCat.coe_comp