state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =
SheafedSpace... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _ | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this✝ :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =
SheafedSpac... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | infer_instance | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier) =
(imageBasicOpen f g U s).carrier | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1) | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case e
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ⇑f.val.base ≫ ⇑(coequalizer.π f.val g.val).base = ⇑g.val.base ≫ ⇑(coequalizer.π f.val g.val).base | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | ext | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case e.h
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
a✝ : (forget TopCat).obj ↑X.toPresheafedSpace
⊢ (⇑f.val.base ≫ ⇑(coequalizer.π f.val g.val).base) a✝ = (⇑g.val.base ≫ ⇑(coequalizer.π f.val g.val).... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base] | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case e.h
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
a✝ : (forget TopCat).obj ↑X.toPresheafedSpace
⊢ (f.val ≫ coequalizer.π f.val g.val).base a✝ = (g.val ≫ coequalizer.π f.val g.val).base a✝ | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | congr 2 | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case e.h.e_a.e_self
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
a✝ : (forget TopCat).obj ↑X.toPresheafedSpace
⊢ f.val ≫ coequalizer.π f.val g.val = g.val ≫ coequalizer.π f.val g.val | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact coequalizer.condition f.1 g.1 | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case h
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsColimit
(Cofork.ofπ ⇑(coequalizer.π f.val g.val).base
(_ : ⇑f.val.base ≫ ⇑(coequalizer.π f.val g.val).base = ⇑g.val.base ≫ ⇑(coequalize... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply isColimitCoforkMapOfIsColimit (forget TopCat) | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case h.l
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsColimit (Cofork.ofπ (coequalizer.π f.val g.val).base ?h.w)
case h.w
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _) | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case h.l.l
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsColimit (Cofork.ofπ (coequalizer.π f.val g.val) ?h.l.w)
case h.l.w
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val)... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact coequalizerIsCoequalizer f.1 g.1 | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ⇑f.val.base ⁻¹' (imageBasicOpen f g U s).carrier = ⇑g.val.base ⁻¹' (imageBasicOpen f g U s).carrier | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
this : (Opens.map f.val.base).obj (imageBasicOpen f g U s) = (Opens.map g.val.base).obj (imageBasicOpen f g U s)
⊢ ⇑f.val.base ⁻¹' (imageBasicOpen f g U s).... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | injection this | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ (Opens.map f.val.base).obj (imageBasicOpen f g U s) = (Opens.map g.val.base).obj (imageBasicOpen f g U s) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | delta imageBasicOpen | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ (Opens.map f.val.base).obj
(RingedSpace.basicOpen (toRingedSpace Y)
(let_fun this := ((coequalizer.π f.val g.val).c.app (op U)) s;
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [preimage_basicOpen f, preimage_basicOpen g] | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ RingedSpace.basicOpen (toRingedSpace X)
((f.val.c.app
{
unop :=
{
unop :=
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | dsimp only [Functor.op, unop_op] | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ RingedSpace.basicOpen (toRingedSpace X)
((f.val.c.app
{
unop :=
{ carrier := ⇑(coequalizer.π f.val g.val)... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res] | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ((Opens.map (f.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)).unop ⊓
RingedSpace.basicOpen (toRingedSpace X) (((g.val ≫ coequalizer.π ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply inf_eq_right.mpr | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ RingedSpace.basicOpen (toRingedSpace X) (((g.val ≫ coequalizer.π f.val g.val).c.app (op U)) s) ≤
((Opens.map (f.val ≫ coequalizer.π f.val g.val... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | refine' (RingedSpace.basicOpen_le _ _).trans _ | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ((Opens.map (g.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)).unop ≤
((Opens.map (f.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [coequalizer.condition f.1 g.1] | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen (⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp] | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen
(⇑(TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget CommRingCat) f.val g.val)) ∘
⇑(colimit.ι
(parallelPair ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [← coe_comp] | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen
(⇑(colimit.ι
(parallelPair ((SheafedSpace.forget CommRingCat).map f.val) ((SheafedSpace.forget CommRingCat).map g.val))
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison] | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen
(⇑((SheafedSpace.forget CommRingCat).map (coequalizer.π f.val g.val)) ⁻¹'
(⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | dsimp only [SheafedSpace.forget] | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [imageBasicOpen_image_preimage] | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen (imageBasicOpen f g U s).carrier | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact (imageBasicOpen f g U s).2 | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
x : ↑(toTopCat Y)
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val) x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | constructor | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
x : ↑(toTopCat Y)
⊢ ∀ (a : ↑(PresheafedSpace.stalk (coequalizer f.val g.val).toPresheafedSpace ((coequalizer.π f.val g.val).base x))),
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rintro a ha | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
x : ↑(toTopCat Y)
a : ↑(PresheafedSpace.stalk (coequalizer f.val g.val).toPresheafedSpace ((coequalizer.π f.val g.val).base x))
ha : IsUnit... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩ | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | let V := imageBasicOpen f g U s | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _) | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩ | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s] | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply RingHom.isUnit_map | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro.a
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)] | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro.a
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp] | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case map_nonunit.intro.intro.intro.a
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s) | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g | instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
x : ↑↑(Limits.coequalizer f.val g.val).toPresheafedSpace
⊢ LocalRing ↑(TopCat.Presheaf.stalk (Limits.coequalizer f.val g.val).toPresheafedSpace.presheaf x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x | /-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.260_0.tE6q65npbp8AX2g | /-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
y : (forget TopCat).obj ↑Y.toPresheafedSpace
⊢ LocalRing
↑(TopCat.Presheaf.stalk (Limits.coequalizer f.val g.val).toPresheafedSpace.presheaf
((coequalizer.π f.val g.val).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing | /-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.260_0.tE6q65npbp8AX2g | /-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X✝ Y✝ : LocallyRingedSpace
f✝ g✝ : X✝ ⟶ Y✝
X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalRingHom (PresheafedSpace.stalkMap f x)
⊢ IsLocalRingHom (PresheafedSpace.stalkMap g x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x] | theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.278_0.tE6q65npbp8AX2g | theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X✝ Y✝ : LocallyRingedSpace
f✝ g✝ : X✝ ⟶ Y✝
X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalRingHom (PresheafedSpace.stalkMap f x)
⊢ IsLocalRingHom
(eqToHom
(_ :
PresheafedSpace.stalk Y.toPresheafedSpace (g.base x) = PresheafedSpace.stalk Y.toPresheafedSpace (f.base x))... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | infer_instance | theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.278_0.tE6q65npbp8AX2g | theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
⊢ IsColimit (coequalizerCofork f g) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply Cofork.IsColimit.mk' | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create
X Y : LocallyRingedSpace
f g : X ⟶ Y
⊢ (s : Cofork f g) →
{ l //
Cofork.π (coequalizerCofork f g) ≫ l = Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallel... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | intro s | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
⊢ { l //
Cofork.π (coequalizerCofork f g) ≫ l = Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).ob... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
⊢ f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | injection s.condition | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ { l //
Cofork.π (coequalizerCofork f g) ≫ l = Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩ | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_1
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ ∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | intro x | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_1
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩ | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | set h := _ | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ?m.199155 := ?m.199156
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).bas... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | change IsLocalRingHom h | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h) | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _) | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(Presheafe... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y) | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(Presheafe... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | erw [← PresheafedSpace.stalkMap.comp] | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(Presheafe... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(Presheafe... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | infer_instance | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ Cofork.π (coequalizerCofork f g) ≫
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpa... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | constructor | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.left
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ Cofork.π (coequalizerCofork f g) ≫
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace)... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _) | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ ∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingP... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | intro m h | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : Cofork.π (coequalizerCof... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [← h] | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : Cofork.π (coequalizerCof... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rfl | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply LocallyRingedSpace.Hom.ext | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right.val
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.o... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1 | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right.val
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.o... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rintro ⟨⟩ | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right.val.zero
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelP... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc] | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right.val.zero
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelP... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | change _ ≫ _ ≫ _ = _ ≫ _ | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right.val.zero
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelP... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | congr | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case create.refine_2.right.val.one
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPa... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact h | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g | /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ PreservesColimit F forgetToSheafedSpace | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
this :
PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))
forgetToSheafedSpace
⊢ PreservesColimit F forgetToSheafedSpace | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))
forgetToSheafedSpace | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _) | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ IsColimit
(forgetToSheafedSpace.mapCocone
(coequalizerCofork (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | apply (isColimitMapCoconeCoforkEquiv _ _).symm _ | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ IsColimit
(Cofork.ofπ
(forgetToSheafedSpace.map
{ val := coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,
prop :=
(_ :
∀ (x : ↑(toTopCa... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | dsimp only [forgetToSheafedSpace] | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ IsColimit
(Cofork.ofπ (coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val)
(_ :
(F.map WalkingParallelPairHom.left).val ≫
coequalizer.π (F.map WalkingParallelPair... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | exact coequalizerIsCoequalizer _ _ | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair... | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g | noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
h : m ≤ n
⊢ ↑(n - m) + ↑m = ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rw [← cast_add, Nat.sub_add_cancel h] | @[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by | Mathlib.Data.Int.Cast.Basic.32_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
h : 0 < 0
⊢ ↑(0 - 1) = ↑0 - 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | cases h | @[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by | Mathlib.Data.Int.Cast.Basic.38_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
x✝ : 0 < n + 1
⊢ ↑(n + 1 - 1) = ↑(n + 1) - 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rw [cast_succ, add_sub_cancel] | @[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by | Mathlib.Data.Int.Cast.Basic.38_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
x✝ : 0 < n + 1
⊢ ↑(n + 1 - 1) = ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rfl | @[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; | Mathlib.Data.Int.Cast.Basic.38_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝¹ : AddGroupWithOne R
n : ℕ
inst✝ : AtLeastTwo n
⊢ ↑(OfNat.ofNat n) = OfNat.ofNat n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n | @[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
| Mathlib.Data.Int.Cast.Basic.72_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
⊢ ↑1 = 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | erw [cast_ofNat, Nat.cast_one] | @[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
| Mathlib.Data.Int.Cast.Basic.77_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
⊢ ↑(-↑0) = -↑↑0 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | erw [cast_zero, neg_zero] | @[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by | Mathlib.Data.Int.Cast.Basic.83_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg] | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
⊢ ↑(-↑(n + 1)) = -↑↑(n + 1) | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | erw [cast_ofNat, cast_negSucc] | @[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by | Mathlib.Data.Int.Cast.Basic.83_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg] | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
⊢ ↑(- -[n+1]) = -↑-[n+1] | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | erw [cast_ofNat, cast_negSucc, neg_neg] | @[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by | Mathlib.Data.Int.Cast.Basic.83_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg] | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(subNatNat m n) = ↑m - ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | unfold subNatNat | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
| Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(match n - m with
| 0 => ofNat (m - n)
| succ k => -[k+1]) =
↑m - ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | cases e : n - m | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
| Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n | Mathlib_Data_Int_Cast_Basic |
case zero
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
e : n - m = zero
⊢ ↑(match zero with
| 0 => ofNat (m - n)
| succ k => -[k+1]) =
↑m - ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | simp only [ofNat_eq_coe] | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· | Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n | Mathlib_Data_Int_Cast_Basic |
case zero
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
e : n - m = zero
⊢ ↑↑(m - n) = ↑m - ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | simp [e, Nat.le_of_sub_eq_zero e] | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
| Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n | Mathlib_Data_Int_Cast_Basic |
case succ
R : Type u
inst✝ : AddGroupWithOne R
m n n✝ : ℕ
e : n - m = succ n✝
⊢ ↑(match succ n✝ with
| 0 => ofNat (m - n)
| succ k => -[k+1]) =
↑m - ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub] | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· | Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
⊢ ↑(negOfNat n) = -↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | simp [Int.cast_neg, negOfNat_eq] | @[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by | Mathlib.Data.Int.Cast.Basic.104_0.3MsWc9B5PAFbTbn | @[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(↑m + ↑n) = ↑↑m + ↑↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | simp [-Int.natCast_add, ← Int.ofNat_add] | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by | Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_a... | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(↑m + -[n+1]) = ↑↑m + ↑-[n+1] | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg] | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by | Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_a... | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(-[m+1] + ↑n) = ↑-[m+1] + ↑↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm] | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
| Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_a... | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑-[m + n + 1+1] = ↑-[m+1] + ↑-[n+1] | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm] | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_a... | Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_a... | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
m n : ℤ
⊢ ↑(m - n) = ↑m - ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | simp [Int.sub_eq_add_neg, sub_eq_add_neg, Int.cast_neg, Int.cast_add] | @[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n := by
| Mathlib.Data.Int.Cast.Basic.122_0.3MsWc9B5PAFbTbn | @[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℤ
⊢ ↑(bit1 n) = bit1 ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rw [bit1, Int.cast_add, Int.cast_one, cast_bit0] | @[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) :=
by | Mathlib.Data.Int.Cast.Basic.146_0.3MsWc9B5PAFbTbn | @[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) | Mathlib_Data_Int_Cast_Basic |
R : Type u
inst✝ : AddGroupWithOne R
n : ℤ
⊢ bit0 ↑n + 1 = bit1 ↑n | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | rfl | @[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) :=
by rw [bit1, Int.cast_add, Int.cast_one, cast_bit0]; | Mathlib.Data.Int.Cast.Basic.146_0.3MsWc9B5PAFbTbn | @[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) | Mathlib_Data_Int_Cast_Basic |
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