Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.left.single.left.id_single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.fst = s.fst ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.left })))

pp: case refine_2.single.left.pair.left.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op = s.fst ≫ F.presheaf.map (homOfLE ⋯).op

pp: case refine_2.single.left.pair.left.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op = s.fst ≫ F.presheaf.map (homOfLE ⋯).op

pp: case refine_2.single.left.pair.right.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op = s.fst ≫ F.presheaf.map (homOfLE ⋯).op

pp: case refine_2.single.right.single.right.id_single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.snd = s.snd ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.right })))

pp: case refine_2.single.right.pair.left.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op = s.snd ≫ F.presheaf.map (homOfLE ⋯).op

pp: case refine_2.single.right.pair.right.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op = s.snd ≫ F.presheaf.map (homOfLE ⋯).op

pp: case refine_2.single.right.pair.right.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op = s.snd ≫ F.presheaf.map (homOfLE ⋯).op

pp: case refine_2.pair.left.pair.left.id_pair
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op =
    (s.fst ≫ F.obj.map (homOfLE ⋯).op) ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.left } ⊓ ι { down := down✝ })))

pp: case refine_2.pair.right.pair.right.id_pair
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op =
    (s.snd ≫ F.obj.map (homOfLE ⋯).op) ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.right } ⊓ ι { down := down✝ })))

Tactic:
<;>

Goal state after:
case refine_2.single.left.single.left.id_single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.fst = s.fst ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.left })))

case refine_2.single.left.pair.left.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op = s.fst ≫ F.presheaf.map (homOfLE ⋯).op

case refine_2.single.left.pair.left.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op = s.fst ≫ F.presheaf.map (homOfLE ⋯).op

case refine_2.single.left.pair.right.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op = s.fst ≫ F.presheaf.map (homOfLE ⋯).op

case refine_2.single.right.single.right.id_single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.snd = s.snd ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.right })))

case refine_2.single.right.pair.left.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op = s.snd ≫ F.presheaf.map (homOfLE ⋯).op

case refine_2.single.right.pair.right.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op = s.snd ≫ F.presheaf.map (homOfLE ⋯).op

case refine_2.single.right.pair.right.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op = s.snd ≫ F.presheaf.map (homOfLE ⋯).op

case refine_2.pair.left.pair.left.id_pair
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.fst ≫ F.obj.map (homOfLE ⋯).op =
    (s.fst ≫ F.obj.map (homOfLE ⋯).op) ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.left } ⊓ ι { down := down✝ })))

case refine_2.pair.right.pair.right.id_pair
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ 𝟙 s.pt ≫ s.snd ≫ F.obj.map (homOfLE ⋯).op =
    (s.snd ≫ F.obj.map (homOfLE ⋯).op) ≫ F.presheaf.map (𝟙 (op (ι { down := WalkingPair.right } ⊓ ι { down := down✝ })))