Theorem: AlgebraicGeometry.ι_sigmaIsoGlued_inv
File: Mathlib/AlgebraicGeometry/Limits.lean

Proof steps:
state_before: ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ (disjointGlueData f).ι i ≫ (sigmaIsoGlued f).inv = Sigma.ι f i
state_after: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ Scheme.forgetToLocallyRingedSpace.map ((disjointGlueData f).ι i ≫ (sigmaIsoGlued f).inv) =
    Scheme.forgetToLocallyRingedSpace.map (Sigma.ι f i)
tactic: apply Scheme.forgetToLocallyRingedSpace.map_injective
state_before: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ Scheme.forgetToLocallyRingedSpace.map ((disjointGlueData f).ι i ≫ (sigmaIsoGlued f).inv) =
    Scheme.forgetToLocallyRingedSpace.map (Sigma.ι f i)
state_after: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ (disjointGlueData f).ι i ≫
      ((disjointGlueData f).isoLocallyRingedSpace.hom ≫
          (colimit.isoColimitCocone
              { cocone := toLocallyRingedSpaceCoproductCofan f,
                isColimit := toLocallyRingedSpaceCoproductCofanIsColimit f }).inv) ≫
        sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
tactic: dsimp [sigmaIsoGlued]
state_before: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ (disjointGlueData f).ι i ≫
      ((disjointGlueData f).isoLocallyRingedSpace.hom ≫
          (colimit.isoColimitCocone
              { cocone := toLocallyRingedSpaceCoproductCofan f,
                isColimit := toLocallyRingedSpaceCoproductCofanIsColimit f }).inv) ≫
        sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
state_after: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ (disjointGlueData f).ι i ≫
      (disjointGlueData f).isoLocallyRingedSpace.hom ≫
        (colimit.isoColimitCocone
              { cocone := toLocallyRingedSpaceCoproductCofan f,
                isColimit := toLocallyRingedSpaceCoproductCofanIsColimit f }).inv ≫
          sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
tactic: simp only [Category.assoc]
state_before: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ (disjointGlueData f).ι i ≫
      (disjointGlueData f).isoLocallyRingedSpace.hom ≫
        (colimit.isoColimitCocone
              { cocone := toLocallyRingedSpaceCoproductCofan f,
                isColimit := toLocallyRingedSpaceCoproductCofanIsColimit f }).inv ≫
          sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
state_after: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ ((disjointGlueData f).mapGlueData Scheme.forgetToLocallyRingedSpace).ι i ≫
      (colimit.isoColimitCocone
            { cocone := toLocallyRingedSpaceCoproductCofan f,
              isColimit := toLocallyRingedSpaceCoproductCofanIsColimit f }).inv ≫
        sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
tactic: refine ((disjointGlueData f).ι_gluedIso_hom_assoc Scheme.forgetToLocallyRingedSpace i _).trans ?_
state_before: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ ((disjointGlueData f).mapGlueData Scheme.forgetToLocallyRingedSpace).ι i ≫
      (colimit.isoColimitCocone
            { cocone := toLocallyRingedSpaceCoproductCofan f,
              isColimit := toLocallyRingedSpaceCoproductCofanIsColimit f }).inv ≫
        sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
state_after: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ colimit.ι (Discrete.functor (Scheme.toLocallyRingedSpace ∘ f)) { as := i } ≫
      sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
tactic: refine (colimit.isoColimitCocone_ι_inv_assoc
    ⟨_, toLocallyRingedSpaceCoproductCofanIsColimit f⟩ ⟨i⟩ _).trans ?_
state_before: case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ colimit.ι (Discrete.functor (Scheme.toLocallyRingedSpace ∘ f)) { as := i } ≫
      sigmaComparison Scheme.forgetToLocallyRingedSpace f =
    Sigma.ι f i
state_after: no goals
tactic: exact ι_comp_sigmaComparison Scheme.forgetToLocallyRingedSpace _ _