Theorem: CompHausLike.Sigma.openEmbedding_ι
File: Mathlib/Topology/Category/CompHausLike/Limits.lean

Proof steps:
state_before: P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
⊢ OpenEmbedding ⇑(Sigma.ι X a)
state_after: P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
⊢ OpenEmbedding
    (⇑(homeoOfIso ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))) ∘
      ⇑(Sigma.ι X a))
tactic: refine OpenEmbedding.of_comp _ (homeoOfIso ((colimit.isColimit _).coconePointUniqueUpToIso
    (finiteCoproduct.isColimit X))).openEmbedding ?_
state_before: P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
⊢ OpenEmbedding
    (⇑(homeoOfIso ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))) ∘
      ⇑(Sigma.ι X a))
state_after: case h.e'_5.h
P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
e_2✝ : ↑(finiteCoproduct.cofan X).pt.toTop = (forget (CompHausLike P✝)).obj (finiteCoproduct X)
⊢ ⇑(homeoOfIso ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))) ∘
      ⇑(Sigma.ι X a) =
    ⇑(finiteCoproduct.ι X a)
tactic: convert finiteCoproduct.openEmbedding_ι X a
state_before: case h.e'_5.h
P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
e_2✝ : ↑(finiteCoproduct.cofan X).pt.toTop = (forget (CompHausLike P✝)).obj (finiteCoproduct X)
⊢ ⇑(homeoOfIso ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))) ∘
      ⇑(Sigma.ι X a) =
    ⇑(finiteCoproduct.ι X a)
state_after: case h.e'_5.h.h
P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
e_2✝ : ↑(finiteCoproduct.cofan X).pt.toTop = (forget (CompHausLike P✝)).obj (finiteCoproduct X)
x : (forget (CompHausLike P✝)).obj (X a)
⊢ (⇑(homeoOfIso ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))) ∘
        ⇑(Sigma.ι X a))
      x =
    (finiteCoproduct.ι X a) x
tactic: ext x
state_before: case h.e'_5.h.h
P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
e_2✝ : ↑(finiteCoproduct.cofan X).pt.toTop = (forget (CompHausLike P✝)).obj (finiteCoproduct X)
x : (forget (CompHausLike P✝)).obj (X a)
⊢ (⇑(homeoOfIso ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))) ∘
        ⇑(Sigma.ι X a))
      x =
    (finiteCoproduct.ι X a) x
state_after: case h.e'_5.h.h
P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
e_2✝ : ↑(finiteCoproduct.cofan X).pt.toTop = (forget (CompHausLike P✝)).obj (finiteCoproduct X)
x : (forget (CompHausLike P✝)).obj (X a)
⊢ (Sigma.ι X a ≫
        ((compHausLikeToTop P✝).mapIso
            ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))).hom)
      x =
    (finiteCoproduct.ι X a) x
tactic: change (Sigma.ι X a ≫ _) x = _
state_before: case h.e'_5.h.h
P✝ : TopCat → Prop
α : Type w
inst✝² : Finite α
X : α → CompHausLike P✝
inst✝¹ : HasExplicitFiniteCoproduct X
P : TopCat → Prop
inst✝ : HasExplicitFiniteCoproducts P
a : α
e_2✝ : ↑(finiteCoproduct.cofan X).pt.toTop = (forget (CompHausLike P✝)).obj (finiteCoproduct X)
x : (forget (CompHausLike P✝)).obj (X a)
⊢ (Sigma.ι X a ≫
        ((compHausLikeToTop P✝).mapIso
            ((colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))).hom)
      x =
    (finiteCoproduct.ι X a) x
state_after: no goals
tactic: simp