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 |
|---|---|---|---|---|---|---|
case refine'_1.refine'_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ (parallelPair f 0 ⋙ G).map WalkingParallelPairHom.right ≫
(Iso.... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
refine' parallelPair.ext (Iso.refl _) (Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ (Cones.postcompose
(parallelPair.ext (Iso.refl ((parallelPair f 0 ⋙ G... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | exact Cones.ext (Iso.refl _) (by rintro (_|_) <;> aesop_cat) | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
refine' parallelPair.ext (Iso.refl _) (Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ ∀ (j : WalkingParallelPair),
((Cones.postcompose
(parallelPair.ext (Iso.... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rintro (_|_) | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
refine' parallelPair.ext (Iso.refl _) (Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case zero
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ ((Cones.postcompose
(parallelPair.ext (Iso.refl ((parallelPair f 0 ⋙... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | aesop_cat | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
refine' parallelPair.ext (Iso.refl _) (Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case one
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ ((Cones.postcompose
(parallelPair.ext (Iso.refl ((parallelPair f 0 ⋙ ... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | aesop_cat | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
refine' parallelPair.ext (Iso.refl _) (Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
⊢ G.map h ≫ G.map f = 0 | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp only [← G.map_comp, w, Functor.map_zero] | /-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
This essentially lets us commute `KernelFork.ofι` with `Functor.mapCone`.
This is a variant of `isLimitMapConeForkEquiv` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitiona... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.73_0.Ox2DGCW1z12SA2j | /-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
This essentially lets us commute `KernelFork.ofι` with `Functor.mapCone`.
This is a variant of `isLimitMapConeForkEquiv` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitiona... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : HasZeroMorphisms C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : HasZeroMorphisms D
G : C ⥤ D
inst✝¹ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝ : PreservesLimit (parallelPair f 0) G
l : IsLimit (KernelFork.ofι h w)
⊢ G.map h... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp only [← G.map_comp, w, Functor.map_zero] | /-- The property of preserving kernels expressed in terms of kernel forks.
This is a variant of `isLimitForkMapOfIsLimit` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitionally.
-/
def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G]
(l : IsLimit (KernelFork.of... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.88_0.Ox2DGCW1z12SA2j | /-- The property of preserving kernels expressed in terms of kernel forks.
This is a variant of `isLimitForkMapOfIsLimit` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitionally.
-/
def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G]
(l : IsLimit (KernelFork.of... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝¹ : HasKernel f
inst✝ : PreservesLimit (parallelPair f 0) G
⊢ G.map (kernel.ι f) ... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp only [← G.map_comp, kernel.condition, comp_zero, Functor.map_zero] | /-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit.
-/
def isLimitOfHasKernelOfPreservesLimit [PreservesLimit (parallelPair f 0) G] :
IsLimit
(Fork.ofι (G.map (kernel.ι f))
(by | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.103_0.Ox2DGCW1z12SA2j | /-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit.
-/
def isLimitOfHasKernelOfPreservesLimit [PreservesLimit (parallelPair f 0) G] :
IsLimit
(Fork.ofι (G.map (kernel.ι f))
(by simp only [← G.map_comp, kernel.condition, comp_z... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝¹ : HasKernel f
inst✝ : HasKernel (G.map f)
i : IsIso (kernelComparison f G)
⊢ Pr... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | apply preservesLimitOfPreservesLimitCone (kernelIsKernel f) | /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
PreservesLimit (parallelPair f 0) G := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.119_0.Ox2DGCW1z12SA2j | /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
PreservesLimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝¹ : HasKernel f
inst✝ : HasKernel (G.map f)
i : IsIso (kernelComparison f G)
⊢ Is... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | apply (isLimitMapConeForkEquiv' G (kernel.condition f)).symm _ | /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
PreservesLimit (parallelPair f 0) G := by
apply preservesLimitOfPreservesLimitCone (kernelIsKernel f)
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.119_0.Ox2DGCW1z12SA2j | /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
PreservesLimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝¹ : HasKernel f
inst✝ : HasKernel (G.map f)
i : IsIso (kernelComparison f G)
⊢ Is... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | exact @IsLimit.ofPointIso _ _ _ _ _ _ _ (kernelIsKernel (G.map f)) i | /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
PreservesLimit (parallelPair f 0) G := by
apply preservesLimitOfPreservesLimitCone (kernelIsKernel f)
apply (isLimitMapConeForkEqui... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.119_0.Ox2DGCW1z12SA2j | /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
PreservesLimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝² : HasKernel f
inst✝¹ : HasKernel (G.map f)
inst✝ : PreservesLimit (parallelPair... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← cancel_mono (kernel.ι _)] | @[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).hom = kernelComparison f G := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.138_0.Ox2DGCW1z12SA2j | @[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).hom = kernelComparison f G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝² : HasKernel f
inst✝¹ : HasKernel (G.map f)
inst✝ : PreservesLimit (parallelPair... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp [PreservesKernel.iso] | @[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).hom = kernelComparison f G := by
rw [← cancel_mono (kernel.ι _)]
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.138_0.Ox2DGCW1z12SA2j | @[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).hom = kernelComparison f G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝² : HasKernel f
inst✝¹ : HasKernel (G.map f)
inst✝ : PreservesLimit (parallelPair... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← PreservesKernel.iso_hom] | instance : IsIso (kernelComparison f G) := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.144_0.Ox2DGCW1z12SA2j | instance : IsIso (kernelComparison f G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝² : HasKernel f
inst✝¹ : HasKernel (G.map f)
inst✝ : PreservesLimit (parallelPair... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | infer_instance | instance : IsIso (kernelComparison f G) := by
rw [← PreservesKernel.iso_hom]
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.144_0.Ox2DGCW1z12SA2j | instance : IsIso (kernelComparison f G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : HasZeroMorphisms C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : HasZeroMorphisms D
G : C ⥤ D
inst✝⁶ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝⁵ : HasKernel f
inst✝⁴ : HasKernel (G.map f)
inst✝³ : PreservesLimit (parallelPa... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← G.map_comp, hpq, G.map_comp] | @[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
kernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.148_0.Ox2DGCW1z12SA2j | @[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
kernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by rw [← G.map_comp, hpq, G.map_comp]) ≫
... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : HasZeroMorphisms C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : HasZeroMorphisms D
G : C ⥤ D
inst✝⁶ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝⁵ : HasKernel f
inst✝⁴ : HasKernel (G.map f)
inst✝³ : PreservesLimit (parallelPa... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [Iso.comp_inv_eq, Category.assoc, PreservesKernel.iso_hom, Iso.eq_inv_comp,
PreservesKernel.iso_hom, kernelComparison_comp_kernel_map] | @[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
kernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by rw [← G.map_comp, hpq, G.map_comp]) ≫
... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.148_0.Ox2DGCW1z12SA2j | @[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
kernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by rw [← G.map_comp, hpq, G.map_comp]) ≫
... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ G.map f ≫ G.map (Cofork.π c) = 0 | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← G.map_comp, c.condition, G.map_zero] | @[reassoc (attr := simp)]
lemma map_condition : G.map f ≫ G.map c.π = 0 := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.166_0.Ox2DGCW1z12SA2j | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ IsColimit (G.mapCocone c) ≃ IsColimit (map c G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _) | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ parallelPair (G.map f) 0 ≅ parallelPair f 0 ⋙ G | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' parallelPair.ext (Iso.refl _) (Iso.refl _) _ _ | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_1.refine'_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ (parallelPair (G.map f) 0).map WalkingParallelPairHom.left ≫
... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
refine' pa... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_1.refine'_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ (parallelPair (G.map f) 0).map WalkingParallelPairHom.right ≫
... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
refine' pa... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ (Cocones.precompose
(parallelPair.ext (Iso.refl ((parallelPair (G... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | exact Cocones.ext (Iso.refl _) (by rintro (_|_) <;> aesop_cat) | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
refine' pa... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ ∀ (j : WalkingParallelPair),
((Cocones.precompose
(parallelPair.ex... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rintro (_|_) | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
refine' pa... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case zero
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ ((Cocones.precompose
(parallelPair.ext (Iso.refl ((parallelPai... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | aesop_cat | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
refine' pa... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case one
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : CokernelCofork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ ((Cocones.precompose
(parallelPair.ext (Iso.refl ((parallelPair... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | aesop_cat | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
refine' pa... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.178_0.Ox2DGCW1z12SA2j | /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
the mapped cokernel cofork is colimit. -/
def isColimitMapCoconeEquiv :
IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
⊢ G.map f ≫ G.map h = 0 | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp only [← G.map_comp, w, Functor.map_zero] | /-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `CokernelCofork.ofπ` with `Functor.mapCocone`.
This is a variant of `isColimitMapCoconeCoforkEquiv` for equalizers,
which we can't use directly between `G.map 0 = 0` d... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.200_0.Ox2DGCW1z12SA2j | /-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `CokernelCofork.ofπ` with `Functor.mapCocone`.
This is a variant of `isColimitMapCoconeCoforkEquiv` for equalizers,
which we can't use directly between `G.map 0 = 0` d... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : HasZeroMorphisms C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : HasZeroMorphisms D
G : C ⥤ D
inst✝¹ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝ : PreservesColimit (parallelPair f 0) G
l : IsColimit (CokernelCofork.ofπ h w)
⊢... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp only [← G.map_comp, w, Functor.map_zero] | /-- The property of preserving cokernels expressed in terms of cokernel coforks.
This is a variant of `isColimitCoforkMapOfIsColimit` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitionally.
-/
def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G]
(l : Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.215_0.Ox2DGCW1z12SA2j | /-- The property of preserving cokernels expressed in terms of cokernel coforks.
This is a variant of `isColimitCoforkMapOfIsColimit` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitionally.
-/
def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G]
(l : Is... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝¹ : HasCokernel f
inst✝ : PreservesColimit (parallelPair f 0) G
⊢ G.map f ≫ G.map... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp only [← G.map_comp, cokernel.condition, zero_comp, Functor.map_zero] | /--
If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
a cokernel cofork is a colimit.
-/
def isColimitOfHasCokernelOfPreservesColimit [PreservesColimit (parallelPair f 0) G] :
IsColimit
(Cofork.ofπ (G.map (cokernel.π f))
(by | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.230_0.Ox2DGCW1z12SA2j | /--
If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
a cokernel cofork is a colimit.
-/
def isColimitOfHasCokernelOfPreservesColimit [PreservesColimit (parallelPair f 0) G] :
IsColimit
(Cofork.ofπ (G.map (cokernel.π f))
(by simp only [← G.map_comp, ... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝¹ : HasCokernel f
inst✝ : HasCokernel (G.map f)
i : IsIso (cokernelComparison f G... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | apply preservesColimitOfPreservesColimitCocone (cokernelIsCokernel f) | /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
PreservesColimit (parallelPair f 0) G := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.247_0.Ox2DGCW1z12SA2j | /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
PreservesColimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝¹ : HasCokernel f
inst✝ : HasCokernel (G.map f)
i : IsIso (cokernelComparison f G... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | apply (isColimitMapCoconeCoforkEquiv' G (cokernel.condition f)).symm _ | /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
PreservesColimit (parallelPair f 0) G := by
apply preservesColimitOfPreservesColimitCocone (cokernelIsCokernel f)
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.247_0.Ox2DGCW1z12SA2j | /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
PreservesColimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝¹ : HasCokernel f
inst✝ : HasCokernel (G.map f)
i : IsIso (cokernelComparison f G... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | exact @IsColimit.ofPointIso _ _ _ _ _ _ _ (cokernelIsCokernel (G.map f)) i | /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
PreservesColimit (parallelPair f 0) G := by
apply preservesColimitOfPreservesColimitCocone (cokernelIsCokernel f)
apply (is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.247_0.Ox2DGCW1z12SA2j | /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
PreservesColimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝² : HasCokernel f
inst✝¹ : HasCokernel (G.map f)
inst✝ : PreservesColimit (parall... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← cancel_epi (cokernel.π _)] | @[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.267_0.Ox2DGCW1z12SA2j | @[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝² : HasCokernel f
inst✝¹ : HasCokernel (G.map f)
inst✝ : PreservesColimit (parall... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp [PreservesCokernel.iso] | @[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G := by
rw [← cancel_epi (cokernel.π _)]
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.267_0.Ox2DGCW1z12SA2j | @[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝² : HasCokernel f
inst✝¹ : HasCokernel (G.map f)
inst✝ : PreservesColimit (parall... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← PreservesCokernel.iso_inv] | instance : IsIso (cokernelComparison f G) := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.273_0.Ox2DGCW1z12SA2j | instance : IsIso (cokernelComparison f G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝² : HasCokernel f
inst✝¹ : HasCokernel (G.map f)
inst✝ : PreservesColimit (parall... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | infer_instance | instance : IsIso (cokernelComparison f G) := by
rw [← PreservesCokernel.iso_inv]
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.273_0.Ox2DGCW1z12SA2j | instance : IsIso (cokernelComparison f G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : HasZeroMorphisms C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : HasZeroMorphisms D
G : C ⥤ D
inst✝⁶ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝⁵ : HasCokernel f
inst✝⁴ : HasCokernel (G.map f)
inst✝³ : PreservesColimit (para... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← G.map_comp, hpq, G.map_comp] | @[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
(PreservesCokernel.iso G _).hom ≫
cokernel.map (G.map f) (G.map g) (G.map p) (G.map ... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.277_0.Ox2DGCW1z12SA2j | @[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
(PreservesCokernel.iso G _).hom ≫
cokernel.map (G.map f) (G.map g) (G.map p) (G.map ... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : HasZeroMorphisms C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : HasZeroMorphisms D
G : C ⥤ D
inst✝⁶ : Functor.PreservesZeroMorphisms G
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = 0
inst✝⁵ : HasCokernel f
inst✝⁴ : HasCokernel (G.map f)
inst✝³ : PreservesColimit (para... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← Iso.comp_inv_eq, Category.assoc, ← Iso.eq_inv_comp, PreservesCokernel.iso_inv,
cokernel_map_comp_cokernelComparison, PreservesCokernel.iso_inv] | @[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
(PreservesCokernel.iso G _).hom ≫
cokernel.map (G.map f) (G.map g) (G.map p) (G.map ... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.277_0.Ox2DGCW1z12SA2j | @[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
(PreservesCokernel.iso G _).hom ≫
cokernel.map (G.map f) (G.map g) (G.map p) (G.map ... | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cone (parallelPair 0 0)
hc : IsLimit c
⊢ IsLimit (G.mapCone c) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | have := KernelFork.IsLimit.isIso_ι c hc rfl | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.293_0.Ox2DGCW1z12SA2j | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cone (parallelPair 0 0)
hc : IsLimit c
this : IsIso (Fork.ι c)
⊢ IsLimit (G.mapCone c) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' (KernelFork.isLimitMapConeEquiv c G).symm _ | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := KernelFork.IsLimit.isIso_ι c hc rfl
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.293_0.Ox2DGCW1z12SA2j | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cone (parallelPair 0 0)
hc : IsLimit c
this : IsIso (Fork.ι c)
⊢ IsLimit (KernelFork.map c G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' IsLimit.ofIsoLimit (KernelFork.IsLimit.ofId _ (G.map_zero _ _)) _ | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := KernelFork.IsLimit.isIso_ι c hc rfl
refine' (KernelFork.isLimitMapConeEquiv c G).symm _
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.293_0.Ox2DGCW1z12SA2j | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cone (parallelPair 0 0)
hc : IsLimit c
this : IsIso (Fork.ι c)
⊢ KernelFork.ofι (𝟙 (G.obj X)) (_ : 𝟙 (G.obj X) ≫ G.ma... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | exact (Fork.ext (G.mapIso (asIso (Fork.ι c))).symm (by simp)) | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := KernelFork.IsLimit.isIso_ι c hc rfl
refine' (KernelFork.isLimitMapConeEquiv c G).symm _
refine' IsLimit.ofIsoLimit (KernelFork.IsLimit.ofId _ (G.map_zero _ _)) _
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.293_0.Ox2DGCW1z12SA2j | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cone (parallelPair 0 0)
hc : IsLimit c
this : IsIso (Fork.ι c)
⊢ (G.mapIso (asIso (Fork.ι c))).symm.hom ≫ Fork.ι (Kerne... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := KernelFork.IsLimit.isIso_ι c hc rfl
refine' (KernelFork.isLimitMapConeEquiv c G).symm _
refine' IsLimit.ofIsoLimit (KernelFork.IsLimit.ofId _ (G.map_zero _ _)) _
exact (F... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.293_0.Ox2DGCW1z12SA2j | noncomputable instance preservesKernelZero :
PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cocone (parallelPair 0 0)
hc : IsColimit c
⊢ IsColimit (G.mapCocone c) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | have := CokernelCofork.IsColimit.isIso_π c hc rfl | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.301_0.Ox2DGCW1z12SA2j | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cocone (parallelPair 0 0)
hc : IsColimit c
this : IsIso (Cofork.π c)
⊢ IsColimit (G.mapCocone c) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' (CokernelCofork.isColimitMapCoconeEquiv c G).symm _ | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := CokernelCofork.IsColimit.isIso_π c hc rfl
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.301_0.Ox2DGCW1z12SA2j | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cocone (parallelPair 0 0)
hc : IsColimit c
this : IsIso (Cofork.π c)
⊢ IsColimit (CokernelCofork.map c G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' IsColimit.ofIsoColimit (CokernelCofork.IsColimit.ofId _ (G.map_zero _ _)) _ | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := CokernelCofork.IsColimit.isIso_π c hc rfl
refine' (CokernelCofork.isColimitMapCoconeEquiv c G).symm _
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.301_0.Ox2DGCW1z12SA2j | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cocone (parallelPair 0 0)
hc : IsColimit c
this : IsIso (Cofork.π c)
⊢ CokernelCofork.ofπ (𝟙 (G.obj Y)) (_ : G.map 0 ≫... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | exact (Cofork.ext (G.mapIso (asIso (Cofork.π c))) (by simp)) | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := CokernelCofork.IsColimit.isIso_π c hc rfl
refine' (CokernelCofork.isColimitMapCoconeEquiv c G).symm _
refine' IsColimit.ofIsoColimit (CokernelCofork.IsColimit.ofId _ (G.m... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.301_0.Ox2DGCW1z12SA2j | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
c : Cocone (parallelPair 0 0)
hc : IsColimit c
this : IsIso (Cofork.π c)
⊢ Cofork.π (CokernelCofork.ofπ (𝟙 (G.obj Y)) (_ :... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc := by
have := CokernelCofork.IsColimit.isIso_π c hc rfl
refine' (CokernelCofork.isColimitMapCoconeEquiv c G).symm _
refine' IsColimit.ofIsoColimit (CokernelCofork.IsColimit.ofId _ (G.m... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.301_0.Ox2DGCW1z12SA2j | noncomputable instance preservesCokernelZero :
PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
preserves {c} hc | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
f : X ⟶ Y
hf : f = 0
⊢ PreservesLimit (parallelPair f 0) G | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [hf] | /-- The kernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesKernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesLimit (parallelPair f 0) G := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.311_0.Ox2DGCW1z12SA2j | /-- The kernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesKernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesLimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
f : X ⟶ Y
hf : f = 0
⊢ PreservesLimit (parallelPair 0 0) G | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | infer_instance | /-- The kernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesKernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesLimit (parallelPair f 0) G := by
rw [hf]
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.311_0.Ox2DGCW1z12SA2j | /-- The kernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesKernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesLimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
f : X ⟶ Y
hf : f = 0
⊢ PreservesColimit (parallelPair f 0) G | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [hf] | /-- The cokernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesCokernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesColimit (parallelPair f 0) G := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.317_0.Ox2DGCW1z12SA2j | /-- The cokernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesCokernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesColimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
f : X ⟶ Y
hf : f = 0
⊢ PreservesColimit (parallelPair 0 0) G | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | infer_instance | /-- The cokernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesCokernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesColimit (parallelPair f 0) G := by
rw [hf]
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.317_0.Ox2DGCW1z12SA2j | /-- The cokernel of a zero map is preserved by any functor which preserves zero morphisms. -/
noncomputable def preservesCokernelZero' (f : X ⟶ Y) (hf : f = 0) :
PreservesColimit (parallelPair f 0) G | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α α' : Sort u
β : α → Sort v
β' : α' → Sort v
f : (a : α) → β a
f' : (a : α') → β' a
hα : α = α'
h : ∀ (a : α) (a' : α'), HEq a a' → HEq (f a) (f' a')
⊢ HEq f f' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | subst hα | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
| Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
⊢ HEq f f' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a) | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
| Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
⊢ HEq f f' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | have : β = β' := by funext a
exact type_eq_of_heq (this a) | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
| Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
⊢ β = β' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | funext a | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by | Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
case h
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
a : α
⊢ β a = β' a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | exact type_eq_of_heq (this a) | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
| Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this✝ : ∀ (a : α), HEq (f a) (f' a)
this : β = β'
⊢ HEq f f' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | subst this | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (th... | Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f f' : (a : α) → β a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
⊢ HEq f f' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | apply heq_of_eq | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (th... | Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
case h
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f f' : (a : α) → β a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
⊢ f = f' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | funext a | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (th... | Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
case h.h
α✝ : Sort ?u.774
β✝ : Sort ?u.777
γ : Sort ?u.780
f✝ : α✝ → β✝
α : Sort u
β : α → Sort v
f f' : (a : α) → β a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
a : α
⊢ f a = f' a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | exact eq_of_heq (this a) | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (th... | Mathlib.Logic.Function.Basic.70_0.QX1TCPxnrBJfF8i | lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' | Mathlib_Logic_Function_Basic |
α : Type u_1
β : Type u_2
γ : Sort ?u.1351
f : α → β
inst✝³ : BEq α
inst✝² : LawfulBEq α
inst✝¹ : BEq β
inst✝ : LawfulBEq β
I : Injective f
a b : α
⊢ (f a == f b) = (a == b) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | by_cases h : a == b | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
| Mathlib.Logic.Function.Basic.99_0.QX1TCPxnrBJfF8i | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) | Mathlib_Logic_Function_Basic |
case pos
α : Type u_1
β : Type u_2
γ : Sort ?u.1351
f : α → β
inst✝³ : BEq α
inst✝² : LawfulBEq α
inst✝¹ : BEq β
inst✝ : LawfulBEq β
I : Injective f
a b : α
h : (a == b) = true
⊢ (f a == f b) = (a == b) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simp [h] | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> | Mathlib.Logic.Function.Basic.99_0.QX1TCPxnrBJfF8i | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) | Mathlib_Logic_Function_Basic |
case neg
α : Type u_1
β : Type u_2
γ : Sort ?u.1351
f : α → β
inst✝³ : BEq α
inst✝² : LawfulBEq α
inst✝¹ : BEq β
inst✝ : LawfulBEq β
I : Injective f
a b : α
h : ¬(a == b) = true
⊢ (f a == f b) = (a == b) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simp [h] | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> | Mathlib.Logic.Function.Basic.99_0.QX1TCPxnrBJfF8i | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) | Mathlib_Logic_Function_Basic |
case pos
α : Type u_1
β : Type u_2
γ : Sort ?u.1351
f : α → β
inst✝³ : BEq α
inst✝² : LawfulBEq α
inst✝¹ : BEq β
inst✝ : LawfulBEq β
I : Injective f
a b : α
h : (a == b) = true
⊢ f a = f b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simpa [I.eq_iff] using h | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> | Mathlib.Logic.Function.Basic.99_0.QX1TCPxnrBJfF8i | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) | Mathlib_Logic_Function_Basic |
case neg
α : Type u_1
β : Type u_2
γ : Sort ?u.1351
f : α → β
inst✝³ : BEq α
inst✝² : LawfulBEq α
inst✝¹ : BEq β
inst✝ : LawfulBEq β
I : Injective f
a b : α
h : ¬(a == b) = true
⊢ ¬f a = f b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simpa [I.eq_iff] using h | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> | Mathlib.Logic.Function.Basic.99_0.QX1TCPxnrBJfF8i | theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) | Mathlib_Logic_Function_Basic |
α : Sort u_3
β : Sort u_2
γ : Sort u_1
f : α → β
g : γ → α
I : Injective (f ∘ g)
hg : Surjective g
x y : α
h : f x = f y
⊢ x = y | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | obtain ⟨x, rfl⟩ := hg x | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
| Mathlib.Logic.Function.Basic.135_0.QX1TCPxnrBJfF8i | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f | Mathlib_Logic_Function_Basic |
case intro
α : Sort u_3
β : Sort u_2
γ : Sort u_1
f : α → β
g : γ → α
I : Injective (f ∘ g)
hg : Surjective g
y : α
x : γ
h : f (g x) = f y
⊢ g x = y | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | obtain ⟨y, rfl⟩ := hg y | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
| Mathlib.Logic.Function.Basic.135_0.QX1TCPxnrBJfF8i | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f | Mathlib_Logic_Function_Basic |
case intro.intro
α : Sort u_3
β : Sort u_2
γ : Sort u_1
f : α → β
g : γ → α
I : Injective (f ∘ g)
hg : Surjective g
x y : γ
h : f (g x) = f (g y)
⊢ g x = g y | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | exact congr_arg g (I h) | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
| Mathlib.Logic.Function.Basic.135_0.QX1TCPxnrBJfF8i | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
⊢ Injective fun x => if h : p x t... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | intros x₁ x₂ h | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(fun x => if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | dsimp only at h | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : p x₁ then... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | by_cases h₁ : p x₁ | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case pos
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | by_cases h₂ : p x₂ | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case neg
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | by_cases h₂ : p x₂ | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case pos
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | rw [dif_pos h₁, dif_pos h₂] at h | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case pos
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h₁ : p x₁
h₂ :... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | injection (hf h) | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case neg
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | rw [dif_pos h₁, dif_neg h₂] at h | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case neg
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h₁ : p x₁
h₂ :... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | exact (im_disj h).elim | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case pos
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | rw [dif_neg h₁, dif_pos h₂] at h | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case pos
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h₁ : ¬p x₁
h₂ ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | exact (im_disj h.symm).elim | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case neg
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h :
(if h : ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | rw [dif_neg h₁, dif_neg h₂] at h | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
case neg
α : Sort u_1
β : Sort u_2
γ : Sort ?u.3760
f✝ : α → β
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f { val := x, property := hx } ≠ f' { val := x', property := hx' }
x₁ x₂ : α
h₁ : ¬p x₁
h₂ ... | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | injection (hf' h) | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros... | Mathlib.Logic.Function.Basic.161_0.QX1TCPxnrBJfF8i | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.6972
f : α → β
h : ∀ (g₁ g₂ : β → Prop), g₁ ∘ f = g₂ ∘ f → g₁ = g₂
⊢ Surjective f | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩) | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
| Mathlib.Logic.Function.Basic.247_0.QX1TCPxnrBJfF8i | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.6972
f : α → β
h : (fun y => ∃ x, f x = y) = fun x => True
⊢ Surjective f | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | intro y | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
| Mathlib.Logic.Function.Basic.247_0.QX1TCPxnrBJfF8i | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.6972
f : α → β
h : (fun y => ∃ x, f x = y) = fun x => True
y : β
⊢ ∃ a, f a = y | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | rw [congr_fun h y] | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; | Mathlib.Logic.Function.Basic.247_0.QX1TCPxnrBJfF8i | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.6972
f : α → β
h : (fun y => ∃ x, f x = y) = fun x => True
y : β
⊢ True | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | trivial | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; | Mathlib.Logic.Function.Basic.247_0.QX1TCPxnrBJfF8i | theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.7498
f✝ f : α → β
hf : Bijective f
p : β → Prop
x✝ : ∃! y, p y
y : β
hpy : (fun y => p y) y
hy : ∀ (y_1 : β), (fun y => p y) y_1 → y_1 = y
x : α
hx : f x = y
⊢ (fun x => p (f x)) x | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simpa [hx] | theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by | Mathlib.Logic.Function.Basic.266_0.QX1TCPxnrBJfF8i | theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.7498
f✝ f : α → β
hf : Bijective f
p : β → Prop
x✝ : ∃! x, p (f x)
x : α
hpx : (fun x => p (f x)) x
hx : ∀ (y : α), (fun x => p (f x)) y → y = x
y : β
hy : (fun y => p y) y
z : α
hz : f z = y
⊢ (fun x => p (f x)) z | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simpa [hz] | theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, ... | Mathlib.Logic.Function.Basic.266_0.QX1TCPxnrBJfF8i | theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
⊢ ¬Surjective f | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | intro hf | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
| Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | let T : Type max u v := Sigma f | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
| Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
| Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
x✝ : ∃ a, f a = Set T
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
| Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
case intro
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
case intro
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩ | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
case intro
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
⊢ Injective g | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | intro s t h | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
s t : Set T
h : g s = g t
⊢ s = t | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
s t : Set T
h : g s = g t
this : cast hU (g s).snd = cast hU (g t).snd
⊢ s = t | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | simp only [cast_cast, cast_eq] at this | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
s t : Set T
h : g s = g t
this : s = t
⊢ s = t | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | assumption | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
s t : Set T
h : g s = g t
⊢ cast hU (g s).snd = cast hU (g t).snd | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30... | congr | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
... | Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
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