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 |
|---|---|---|---|---|---|---|
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
h : ∀ (x : E), 0 < 1 + ‖x‖ ^ 2
⊢ ContDiff ℝ n fun x => (sqrt (1 + ‖x‖ ^ 2))⁻¹ | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | refine' ContDiff.inv _ fun x => Real.sqrt_ne_zero'.mpr (h x) | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) := by
suffices ContDiff ℝ n fun x : E => (1 + ‖x‖ ^ 2 : ℝ).sqrt⁻¹ from this.smul contDiff_id
have h : ∀ x : E, (0 : ℝ) < (1 : ℝ) + ‖x‖ ^ 2 := fun x => by positivity
| Mathlib.Analysis.InnerProductSpace.Calculus.380_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
h : ∀ (x : E), 0 < 1 + ‖x‖ ^ 2
⊢ ContDiff ℝ n fun x => sqrt (1 + ‖x‖ ^ 2) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (contDiff_const.add <| contDiff_norm_sq ℝ).sqrt fun x => (h x).ne' | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) := by
suffices ContDiff ℝ n fun x : E => (1 + ‖x‖ ^ 2 : ℝ).sqrt⁻¹ from this.smul contDiff_id
have h : ∀ x : E, (0 : ℝ) < (1 : ℝ) + ‖x‖ ^ 2 := fun x => by positivity
refine' ContDiff.inv _ fun x => Real.sqrt_ne_zero'.mpr (h x)
| Mathlib.Analysis.InnerProductSpace.Calculus.380_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
⊢ ContDiffWithinAt ℝ n (↑(PartialHomeomorph.symm univUnitBall)) (ball 0 1) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | apply ContDiffAt.contDiffWithinAt | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
| Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
case h
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
⊢ ContDiffAt ℝ n (↑(PartialHomeomorph.symm univUnitBall)) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | suffices ContDiffAt ℝ n (fun y : E => (1 - ‖y‖ ^ 2 : ℝ).sqrt⁻¹) y from this.smul contDiffAt_id | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
apply ContDiffAt.contDiffWithinAt
| Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
case h
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
⊢ ContDiffAt ℝ n (fun y => (sqrt (1 - ‖y‖ ^ 2))⁻¹) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa [mem_ball_zero_iff, ← _root_.abs_one, ← abs_norm, ← sq_lt_sq, one_pow, ← sub_pos] at hy | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (1 - ‖y‖ ^ 2 : ℝ).sqrt⁻¹) y from this.smul contDiffAt_id
| Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
⊢ 0 < 1 - ‖y‖ ^ 2 | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rwa [mem_ball_zero_iff, ← _root_.abs_one, ← abs_norm, ← sq_lt_sq, one_pow, ← sub_pos] at hy | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (1 - ‖y‖ ^ 2 : ℝ).sqrt⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
| Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
case h
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
h : 0 < 1 - ‖y‖ ^ 2
⊢ ContDiffAt ℝ n (fun y => (sqrt (1 - ‖y‖ ^ 2))⁻¹) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | refine' ContDiffAt.inv _ (Real.sqrt_ne_zero'.mpr h) | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (1 - ‖y‖ ^ 2 : ℝ).sqrt⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa ... | Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
case h
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
h : 0 < 1 - ‖y‖ ^ 2
⊢ ContDiffAt ℝ n (fun y => sqrt (1 - ‖y‖ ^ 2)) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | refine' (contDiffAt_sqrt h.ne').comp y _ | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (1 - ‖y‖ ^ 2 : ℝ).sqrt⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa ... | Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
case h
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
h : 0 < 1 - ‖y‖ ^ 2
⊢ ContDiffAt ℝ n (fun y => 1 - ‖y‖ ^ 2) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact contDiffAt_const.sub (contDiff_norm_sq ℝ).contDiffAt | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) := fun y hy ↦ by
apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (1 - ‖y‖ ^ 2 : ℝ).sqrt⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa ... | Mathlib.Analysis.InnerProductSpace.Calculus.386_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
⊢ ContDiff ℝ n ↑(univBall c r) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | unfold univBall | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) := by
| Mathlib.Analysis.InnerProductSpace.Calculus.417_0.6FECEGgqdb67QLM | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
⊢ ContDiff ℝ n
↑(if h : 0 < r then
PartialHomeomorph.trans' univUnitBall (unitBallBall c r h)
(_ : univUnitBall.toPartialEquiv.target = univUnitBall.toPartialEquiv.target)
else Homeomorph.toPartialHom... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | split_ifs with h | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) := by
unfold univBall; | Mathlib.Analysis.InnerProductSpace.Calculus.417_0.6FECEGgqdb67QLM | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
case pos
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : 0 < r
⊢ ContDiff ℝ n
↑(PartialHomeomorph.trans' univUnitBall (unitBallBall c r h)
(_ : univUnitBall.toPartialEquiv.target = univUnitBall.toPartialEquiv.target)) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (contDiff_unitBallBall h).comp contDiff_univUnitBall | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) := by
unfold univBall; split_ifs with h
· | Mathlib.Analysis.InnerProductSpace.Calculus.417_0.6FECEGgqdb67QLM | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
case neg
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : ¬0 < r
⊢ ContDiff ℝ n ↑(Homeomorph.toPartialHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c))) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact contDiff_id.add contDiff_const | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) := by
unfold univBall; split_ifs with h
· exact (contDiff_unitBallBall h).comp contDiff_univUnitBall
· | Mathlib.Analysis.InnerProductSpace.Calculus.417_0.6FECEGgqdb67QLM | theorem contDiff_univBall : ContDiff ℝ n (univBall c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
⊢ ContDiffOn ℝ n (↑(PartialHomeomorph.symm (univBall c r))) (ball c r) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | unfold univBall | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) := by
| Mathlib.Analysis.InnerProductSpace.Calculus.422_0.6FECEGgqdb67QLM | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
⊢ ContDiffOn ℝ n
(↑(PartialHomeomorph.symm
(if h : 0 < r then
PartialHomeomorph.trans' univUnitBall (unitBallBall c r h)
(_ : univUnitBall.toPartialEquiv.target = univUnitBall.toPartialEquiv.tar... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | split_ifs with h | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) := by
unfold univBall; | Mathlib.Analysis.InnerProductSpace.Calculus.422_0.6FECEGgqdb67QLM | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
case pos
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : 0 < r
⊢ ContDiffOn ℝ n
(↑(PartialHomeomorph.symm
(PartialHomeomorph.trans' univUnitBall (unitBallBall c r h)
(_ : univUnitBall.toPartialEquiv.target = univUnitBall.toPartialEquiv.target))))
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | refine contDiffOn_univUnitBall_symm.comp (contDiff_unitBallBall_symm h).contDiffOn ?_ | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) := by
unfold univBall; split_ifs with h
· | Mathlib.Analysis.InnerProductSpace.Calculus.422_0.6FECEGgqdb67QLM | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
case pos
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : 0 < r
⊢ ball c r ⊆ (fun x => ↑(PartialEquiv.symm (unitBallBall c r h).toPartialEquiv) x) ⁻¹' ball 0 1 | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← unitBallBall_source c r h, ← unitBallBall_target c r h] | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) := by
unfold univBall; split_ifs with h
· refine contDiffOn_univUnitBall_symm.comp (contDiff_unitBallBall_symm h).contDiffOn ?_
| Mathlib.Analysis.InnerProductSpace.Calculus.422_0.6FECEGgqdb67QLM | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
case pos
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : 0 < r
⊢ (unitBallBall c r h).toPartialEquiv.target ⊆
(fun x => ↑(PartialEquiv.symm (unitBallBall c r h).toPartialEquiv) x) ⁻¹' (unitBallBall c r h).toPartialEquiv.source | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | apply PartialHomeomorph.symm_mapsTo | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) := by
unfold univBall; split_ifs with h
· refine contDiffOn_univUnitBall_symm.comp (contDiff_unitBallBall_symm h).contDiffOn ?_
rw [← unitBallBall_source c r h, ← unitBallBall_target c r h]
| Mathlib.Analysis.InnerProductSpace.Calculus.422_0.6FECEGgqdb67QLM | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
case neg
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : ¬0 < r
⊢ ContDiffOn ℝ n
(↑(PartialHomeomorph.symm
(Homeomorph.toPartialHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c)))))
(ball c r) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact contDiffOn_id.sub contDiffOn_const | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) := by
unfold univBall; split_ifs with h
· refine contDiffOn_univUnitBall_symm.comp (contDiff_unitBallBall_symm h).contDiffOn ?_
rw [← unitBallBall_source c r h, ← unitBallBall_target c r h]
apply PartialHomeomorph.symm_maps... | Mathlib.Analysis.InnerProductSpace.Calculus.422_0.6FECEGgqdb67QLM | theorem contDiffOn_univBall_symm :
ContDiffOn ℝ n (univBall c r).symm (ball c r) | Mathlib_Analysis_InnerProductSpace_Calculus |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A B : Subspace K V
⊢ A.carrier = B.carrier → A = B | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | cases A | instance : SetLike (Subspace K V) (ℙ K V) where
coe := carrier
coe_injective' A B := by
| Mathlib.LinearAlgebra.Projectivization.Subspace.57_0.Kl46zKj9ofblwLe | instance : SetLike (Subspace K V) (ℙ K V) where
coe | Mathlib_LinearAlgebra_Projectivization_Subspace |
case mk
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B : Subspace K V
carrier✝ : Set (ℙ K V)
mem_add'✝ :
∀ (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) (hvw : v + w ≠ 0),
Projectivization.mk K v hv ∈ carrier✝ →
Projectivization.mk K w hw ∈ carrier✝ → Projectivization.mk K (v ... | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | cases B | instance : SetLike (Subspace K V) (ℙ K V) where
coe := carrier
coe_injective' A B := by
cases A
| Mathlib.LinearAlgebra.Projectivization.Subspace.57_0.Kl46zKj9ofblwLe | instance : SetLike (Subspace K V) (ℙ K V) where
coe | Mathlib_LinearAlgebra_Projectivization_Subspace |
case mk.mk
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
carrier✝¹ : Set (ℙ K V)
mem_add'✝¹ :
∀ (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) (hvw : v + w ≠ 0),
Projectivization.mk K v hv ∈ carrier✝¹ →
Projectivization.mk K w hw ∈ carrier✝¹ → Projectivization.mk K (v + w) hvw ∈... | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | simp | instance : SetLike (Subspace K V) (ℙ K V) where
coe := carrier
coe_injective' A B := by
cases A
cases B
| Mathlib.LinearAlgebra.Projectivization.Subspace.57_0.Kl46zKj9ofblwLe | instance : SetLike (Subspace K V) (ℙ K V) where
coe | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (ℙ K V)
B : Subspace K V
⊢ A ≤ ↑B → span A ≤ B | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | intro h x hx | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS := span S
gc A B :=
⟨fun h => le_trans (subset_span _) h, by
| Mathlib.LinearAlgebra.Projectivization.Subspace.95_0.Kl46zKj9ofblwLe | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (ℙ K V)
B : Subspace K V
h : A ≤ ↑B
x : ℙ K V
hx : x ∈ span A
⊢ x ∈ B | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | induction' hx with y hy | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS := span S
gc A B :=
⟨fun h => le_trans (subset_span _) h, by
intro h x hx... | Mathlib.LinearAlgebra.Projectivization.Subspace.95_0.Kl46zKj9ofblwLe | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS | Mathlib_LinearAlgebra_Projectivization_Subspace |
case of
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (ℙ K V)
B : Subspace K V
h : A ≤ ↑B
x y : ℙ K V
hy : y ∈ A
⊢ y ∈ B | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | apply h | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS := span S
gc A B :=
⟨fun h => le_trans (subset_span _) h, by
intro h x hx... | Mathlib.LinearAlgebra.Projectivization.Subspace.95_0.Kl46zKj9ofblwLe | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS | Mathlib_LinearAlgebra_Projectivization_Subspace |
case of.a
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (ℙ K V)
B : Subspace K V
h : A ≤ ↑B
x y : ℙ K V
hy : y ∈ A
⊢ y ∈ A | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | assumption | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS := span S
gc A B :=
⟨fun h => le_trans (subset_span _) h, by
intro h x hx... | Mathlib.LinearAlgebra.Projectivization.Subspace.95_0.Kl46zKj9ofblwLe | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS | Mathlib_LinearAlgebra_Projectivization_Subspace |
case mem_add
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (ℙ K V)
B : Subspace K V
h : A ≤ ↑B
x : ℙ K V
v✝ w✝ : V
hv✝ : v✝ ≠ 0
hw✝ : w✝ ≠ 0
hvw✝ : v✝ + w✝ ≠ 0
a✝¹ : spanCarrier A (Projectivization.mk K v✝ hv✝)
a✝ : spanCarrier A (Projectivization.mk K w✝ hw✝)
a_ih✝¹ : Pr... | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | apply B.mem_add | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS := span S
gc A B :=
⟨fun h => le_trans (subset_span _) h, by
intro h x hx... | Mathlib.LinearAlgebra.Projectivization.Subspace.95_0.Kl46zKj9ofblwLe | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS | Mathlib_LinearAlgebra_Projectivization_Subspace |
case mem_add.a
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (ℙ K V)
B : Subspace K V
h : A ≤ ↑B
x : ℙ K V
v✝ w✝ : V
hv✝ : v✝ ≠ 0
hw✝ : w✝ ≠ 0
hvw✝ : v✝ + w✝ ≠ 0
a✝¹ : spanCarrier A (Projectivization.mk K v✝ hv✝)
a✝ : spanCarrier A (Projectivization.mk K w✝ hw✝)
a_ih✝¹ : ... | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | assumption' | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS := span S
gc A B :=
⟨fun h => le_trans (subset_span _) h, by
intro h x hx... | Mathlib.LinearAlgebra.Projectivization.Subspace.95_0.Kl46zKj9ofblwLe | /-- The span of a set of points is a Galois insertion between sets of points of a projective space
and subspaces of the projective space. -/
def gi : GaloisInsertion (span : Set (ℙ K V) → Subspace K V) SetLike.coe where
choice S _hS | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (Subspace K V)
v w : V
hv : v ≠ 0
hw : w ≠ 0
hvw : v + w ≠ 0
h1 : Projectivization.mk K v hv ∈ sInf (SetLike.coe '' A)
h2 : Projectivization.mk K w hw ∈ sInf (SetLike.coe '' A)
t : Set (ℙ K V)
⊢ t ∈ SetLike.coe '' A → Projecti... | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | rintro ⟨s, hs, rfl⟩ | /-- Infimums of arbitrary collections of subspaces exist. -/
instance instInfSet : InfSet (Subspace K V) :=
⟨fun A =>
⟨sInf (SetLike.coe '' A), fun v w hv hw hvw h1 h2 t => by
| Mathlib.LinearAlgebra.Projectivization.Subspace.126_0.Kl46zKj9ofblwLe | /-- Infimums of arbitrary collections of subspaces exist. -/
instance instInfSet : InfSet (Subspace K V) | Mathlib_LinearAlgebra_Projectivization_Subspace |
case intro.intro
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
A : Set (Subspace K V)
v w : V
hv : v ≠ 0
hw : w ≠ 0
hvw : v + w ≠ 0
h1 : Projectivization.mk K v hv ∈ sInf (SetLike.coe '' A)
h2 : Projectivization.mk K w hw ∈ sInf (SetLike.coe '' A)
s : Subspace K V
hs : s ∈ A
⊢ Pr... | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | exact s.mem_add v w hv hw _ (h1 s ⟨s, hs, rfl⟩) (h2 s ⟨s, hs, rfl⟩) | /-- Infimums of arbitrary collections of subspaces exist. -/
instance instInfSet : InfSet (Subspace K V) :=
⟨fun A =>
⟨sInf (SetLike.coe '' A), fun v w hv hw hvw h1 h2 t => by
rintro ⟨s, hs, rfl⟩
| Mathlib.LinearAlgebra.Projectivization.Subspace.126_0.Kl46zKj9ofblwLe | /-- Infimums of arbitrary collections of subspaces exist. -/
instance instInfSet : InfSet (Subspace K V) | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
src✝ : Inf (Subspace K V) := inferInstance
⊢ ∀ (s : Set (Subspace K V)), IsGLB s (sInf s) | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | refine fun s => ⟨fun a ha x hx => hx _ ⟨a, ha, rfl⟩, fun a ha x hx E => ?_⟩ | /-- The subspaces of a projective space form a complete lattice. -/
instance : CompleteLattice (Subspace K V) :=
{ (inferInstance : Inf (Subspace K V)),
completeLatticeOfInf (Subspace K V)
(by
| Mathlib.LinearAlgebra.Projectivization.Subspace.134_0.Kl46zKj9ofblwLe | /-- The subspaces of a projective space form a complete lattice. -/
instance : CompleteLattice (Subspace K V) | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
src✝ : Inf (Subspace K V) := inferInstance
s : Set (Subspace K V)
a : Subspace K V
ha : a ∈ lowerBounds s
x : ℙ K V
hx : x ∈ a
E : Set (ℙ K V)
⊢ E ∈ SetLike.coe '' s → x ∈ E | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | rintro ⟨E, hE, rfl⟩ | /-- The subspaces of a projective space form a complete lattice. -/
instance : CompleteLattice (Subspace K V) :=
{ (inferInstance : Inf (Subspace K V)),
completeLatticeOfInf (Subspace K V)
(by
refine fun s => ⟨fun a ha x hx => hx _ ⟨a, ha, rfl⟩, fun a ha x hx E => ?_⟩
| Mathlib.LinearAlgebra.Projectivization.Subspace.134_0.Kl46zKj9ofblwLe | /-- The subspaces of a projective space form a complete lattice. -/
instance : CompleteLattice (Subspace K V) | Mathlib_LinearAlgebra_Projectivization_Subspace |
case intro.intro
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
src✝ : Inf (Subspace K V) := inferInstance
s : Set (Subspace K V)
a : Subspace K V
ha : a ∈ lowerBounds s
x : ℙ K V
hx : x ∈ a
E : Subspace K V
hE : E ∈ s
⊢ x ∈ ↑E | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | exact ha hE hx | /-- The subspaces of a projective space form a complete lattice. -/
instance : CompleteLattice (Subspace K V) :=
{ (inferInstance : Inf (Subspace K V)),
completeLatticeOfInf (Subspace K V)
(by
refine fun s => ⟨fun a ha x hx => hx _ ⟨a, ha, rfl⟩, fun a ha x hx E => ?_⟩
rintro ⟨E, hE, rfl⟩
... | Mathlib.LinearAlgebra.Projectivization.Subspace.134_0.Kl46zKj9ofblwLe | /-- The subspaces of a projective space form a complete lattice. -/
instance : CompleteLattice (Subspace K V) | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ span Set.univ = ⊤ | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | rw [eq_top_iff, SetLike.le_def] | /-- The span of the entire projective space is the top of the lattice of subspaces. -/
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ := by
| Mathlib.LinearAlgebra.Projectivization.Subspace.154_0.Kl46zKj9ofblwLe | /-- The span of the entire projective space is the top of the lattice of subspaces. -/
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ ∀ ⦃x : ℙ K V⦄, x ∈ ⊤ → x ∈ span Set.univ | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | intro x _hx | /-- The span of the entire projective space is the top of the lattice of subspaces. -/
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ := by
rw [eq_top_iff, SetLike.le_def]
| Mathlib.LinearAlgebra.Projectivization.Subspace.154_0.Kl46zKj9ofblwLe | /-- The span of the entire projective space is the top of the lattice of subspaces. -/
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : ℙ K V
_hx : x ∈ ⊤
⊢ x ∈ span Set.univ | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | exact subset_span _ (Set.mem_univ x) | /-- The span of the entire projective space is the top of the lattice of subspaces. -/
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ := by
rw [eq_top_iff, SetLike.le_def]
intro x _hx
| Mathlib.LinearAlgebra.Projectivization.Subspace.154_0.Kl46zKj9ofblwLe | /-- The span of the entire projective space is the top of the lattice of subspaces. -/
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
W : Subspace K V
⊢ W ⊔ span S = span (↑W ∪ S) | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | rw [span_union, span_coe] | /-- The supremum of a subspace and the span of a set of points is equal to the span of the union of
the subspace and the set of points. -/
theorem sup_span {S : Set (ℙ K V)} {W : Subspace K V} : W ⊔ span S = span (W ∪ S) := by
| Mathlib.LinearAlgebra.Projectivization.Subspace.191_0.Kl46zKj9ofblwLe | /-- The supremum of a subspace and the span of a set of points is equal to the span of the union of
the subspace and the set of points. -/
theorem sup_span {S : Set (ℙ K V)} {W : Subspace K V} : W ⊔ span S = span (W ∪ S) | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
W : Subspace K V
⊢ span S ⊔ W = span (S ∪ ↑W) | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | rw [span_union, span_coe] | theorem span_sup {S : Set (ℙ K V)} {W : Subspace K V} : span S ⊔ W = span (S ∪ W) := by
| Mathlib.LinearAlgebra.Projectivization.Subspace.197_0.Kl46zKj9ofblwLe | theorem span_sup {S : Set (ℙ K V)} {W : Subspace K V} : span S ⊔ W = span (S ∪ W) | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
u : ℙ K V
⊢ u ∈ span S ↔ ∀ (W : Subspace K V), S ⊆ ↑W → u ∈ W | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | simp_rw [← span_le_subspace_iff] | /-- A point in a projective space is contained in the span of a set of points if and only if the
point is contained in all subspaces of the projective space which contain the set of points. -/
theorem mem_span {S : Set (ℙ K V)} (u : ℙ K V) :
u ∈ span S ↔ ∀ W : Subspace K V, S ⊆ W → u ∈ W := by
| Mathlib.LinearAlgebra.Projectivization.Subspace.201_0.Kl46zKj9ofblwLe | /-- A point in a projective space is contained in the span of a set of points if and only if the
point is contained in all subspaces of the projective space which contain the set of points. -/
theorem mem_span {S : Set (ℙ K V)} (u : ℙ K V) :
u ∈ span S ↔ ∀ W : Subspace K V, S ⊆ W → u ∈ W | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
u : ℙ K V
⊢ u ∈ span S ↔ ∀ (W : Subspace K V), span S ≤ W → u ∈ W | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | exact ⟨fun hu W hW => hW hu, fun W => W (span S) (le_refl _)⟩ | /-- A point in a projective space is contained in the span of a set of points if and only if the
point is contained in all subspaces of the projective space which contain the set of points. -/
theorem mem_span {S : Set (ℙ K V)} (u : ℙ K V) :
u ∈ span S ↔ ∀ W : Subspace K V, S ⊆ W → u ∈ W := by
simp_rw [← span_le_... | Mathlib.LinearAlgebra.Projectivization.Subspace.201_0.Kl46zKj9ofblwLe | /-- A point in a projective space is contained in the span of a set of points if and only if the
point is contained in all subspaces of the projective space which contain the set of points. -/
theorem mem_span {S : Set (ℙ K V)} (u : ℙ K V) :
u ∈ span S ↔ ∀ W : Subspace K V, S ⊆ W → u ∈ W | Mathlib_LinearAlgebra_Projectivization_Subspace |
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
⊢ span S = sInf {W | S ⊆ ↑W} | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | ext x | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by
| Mathlib.LinearAlgebra.Projectivization.Subspace.209_0.Kl46zKj9ofblwLe | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } | Mathlib_LinearAlgebra_Projectivization_Subspace |
case carrier.h
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
x : ℙ K V
⊢ x ∈ (span S).carrier ↔ x ∈ (sInf {W | S ⊆ ↑W}).carrier | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | simp_rw [mem_carrier_iff, mem_span x] | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by
ext x
| Mathlib.LinearAlgebra.Projectivization.Subspace.209_0.Kl46zKj9ofblwLe | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } | Mathlib_LinearAlgebra_Projectivization_Subspace |
case carrier.h
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
x : ℙ K V
⊢ (∀ (W : Subspace K V), S ⊆ ↑W → x ∈ W) ↔ x ∈ sInf {W | S ⊆ ↑W} | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | refine ⟨fun hx => ?_, fun hx W hW => ?_⟩ | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by
ext x
simp_rw [mem_carrier_iff, mem_span x]
| Mathlib.LinearAlgebra.Projectivization.Subspace.209_0.Kl46zKj9ofblwLe | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } | Mathlib_LinearAlgebra_Projectivization_Subspace |
case carrier.h.refine_1
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
x : ℙ K V
hx : ∀ (W : Subspace K V), S ⊆ ↑W → x ∈ W
⊢ x ∈ sInf {W | S ⊆ ↑W} | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | rintro W ⟨T, hT, rfl⟩ | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by
ext x
simp_rw [mem_carrier_iff, mem_span x]
refine ⟨fun hx => ?_, fun hx W hW => ?_⟩
· | Mathlib.LinearAlgebra.Projectivization.Subspace.209_0.Kl46zKj9ofblwLe | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } | Mathlib_LinearAlgebra_Projectivization_Subspace |
case carrier.h.refine_1.intro.intro
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
x : ℙ K V
hx : ∀ (W : Subspace K V), S ⊆ ↑W → x ∈ W
T : Subspace K V
hT : T ∈ {W | S ⊆ ↑W}
⊢ x ∈ ↑T | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | exact hx T hT | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by
ext x
simp_rw [mem_carrier_iff, mem_span x]
refine ⟨fun hx => ?_, fun hx W hW => ?_⟩
· r... | Mathlib.LinearAlgebra.Projectivization.Subspace.209_0.Kl46zKj9ofblwLe | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } | Mathlib_LinearAlgebra_Projectivization_Subspace |
case carrier.h.refine_2
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set (ℙ K V)
x : ℙ K V
hx : x ∈ sInf {W | S ⊆ ↑W}
W : Subspace K V
hW : S ⊆ ↑W
⊢ x ∈ W | /-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | exact (@sInf_le _ _ { W : Subspace K V | S ⊆ ↑W } W hW) hx | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by
ext x
simp_rw [mem_carrier_iff, mem_span x]
refine ⟨fun hx => ?_, fun hx W hW => ?_⟩
· r... | Mathlib.LinearAlgebra.Projectivization.Subspace.209_0.Kl46zKj9ofblwLe | /-- The span of a set of points in a projective space is equal to the infimum of the collection of
subspaces which contain the set. -/
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } | Mathlib_LinearAlgebra_Projectivization_Subspace |
X Y : WalkingParallelPair
f : WalkingParallelPairHom X Y
⊢ comp f (id Y) = f | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases f | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.101_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left
⊢ comp left (id one) = left | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.101_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right
⊢ comp right (id one) = right | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.101_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id
X : WalkingParallelPair
⊢ comp (id X) (id X) = id X | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.101_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
X Y Z W : WalkingParallelPair
f : WalkingParallelPairHom X Y
g : WalkingParallelPairHom Y Z
h : WalkingParallelPairHom Z W
⊢ comp (comp f g) h = comp f (comp g h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases f | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left
Z W : WalkingParallelPair
h : WalkingParallelPairHom Z W
g : WalkingParallelPairHom one Z
⊢ comp (comp left g) h = comp left (comp g h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases g | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right
Z W : WalkingParallelPair
h : WalkingParallelPairHom Z W
g : WalkingParallelPairHom one Z
⊢ comp (comp right g) h = comp right (comp g h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases g | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id
X Z W : WalkingParallelPair
h : WalkingParallelPairHom Z W
g : WalkingParallelPairHom X Z
⊢ comp (comp (id X) g) h = comp (id X) (comp g h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases g | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left.id
W : WalkingParallelPair
h : WalkingParallelPairHom one W
⊢ comp (comp left (id one)) h = comp left (comp (id one) h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases h | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right.id
W : WalkingParallelPair
h : WalkingParallelPairHom one W
⊢ comp (comp right (id one)) h = comp right (comp (id one) h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases h | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.left
W : WalkingParallelPair
h : WalkingParallelPairHom one W
⊢ comp (comp (id zero) left) h = comp (id zero) (comp left h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases h | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.right
W : WalkingParallelPair
h : WalkingParallelPairHom one W
⊢ comp (comp (id zero) right) h = comp (id zero) (comp right h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases h | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.id
X W : WalkingParallelPair
h : WalkingParallelPairHom X W
⊢ comp (comp (id X) (id X)) h = comp (id X) (comp (id X) h) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases h | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left.id.id
⊢ comp (comp left (id one)) (id one) = comp left (comp (id one) (id one)) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right.id.id
⊢ comp (comp right (id one)) (id one) = comp right (comp (id one) (id one)) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.left.id
⊢ comp (comp (id zero) left) (id one) = comp (id zero) (comp left (id one)) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.right.id
⊢ comp (comp (id zero) right) (id one) = comp (id zero) (comp right (id one)) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.id.left
⊢ comp (comp (id zero) (id zero)) left = comp (id zero) (comp (id zero) left) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.id.right
⊢ comp (comp (id zero) (id zero)) right = comp (id zero) (comp (id zero) right) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.id.id
X : WalkingParallelPair
⊢ comp (comp (id X) (id X)) (id X) = comp (id X) (comp (id X) (id X)) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by
cases f <;> cases g <;> cases h <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.105_0.eJEUq2AFfmN187w | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
X : WalkingParallelPair
⊢ sizeOf (𝟙 X) = 1 + sizeOf X | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases X | @[simp]
theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X := by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.125_0.eJEUq2AFfmN187w | @[simp]
theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case zero
⊢ sizeOf (𝟙 zero) = 1 + sizeOf zero | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | @[simp]
theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X := by cases X <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.125_0.eJEUq2AFfmN187w | @[simp]
theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case one
⊢ sizeOf (𝟙 one) = 1 + sizeOf one | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | @[simp]
theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X := by cases X <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.125_0.eJEUq2AFfmN187w | @[simp]
theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
x : WalkingParallelPair
⊢ WalkingParallelPair | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases x | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case zero
⊢ WalkingParallelPair
case one ⊢ WalkingParallelPair | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exacts [one, zero] | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
X✝ Y✝ : WalkingParallelPair
f : X✝ ⟶ Y✝
⊢ (fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one) (fun h => zero)
(_ : x = x)))
X✝ ⟶
(fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → Walking... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases f | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left
⊢ (fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one) (fun h => zero)
(_ : x = x)))
zero ⟶
(fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => on... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply Quiver.Hom.op | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right
⊢ (fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one) (fun h => zero)
(_ : x = x)))
zero ⟶
(fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => o... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply Quiver.Hom.op | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id
X✝ : WalkingParallelPair
⊢ (fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one) (fun h => zero)
(_ : x = x)))
X✝ ⟶
(fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParalle... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply Quiver.Hom.op | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left.f
⊢ WalkingParallelPair.casesOn (motive := fun t => one = t → WalkingParallelPair) one (fun h => one) (fun h => zero)
(_ : one = one) ⟶
WalkingParallelPair.casesOn (motive := fun t => zero = t → WalkingParallelPair) zero (fun h => one) (fun h => zero)
(_ : zero = zero)
case right.f
⊢ WalkingPa... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exacts [left, right, WalkingParallelPairHom.id _] | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
⊢ ∀ {X Y Z : WalkingParallelPair} (f : X ⟶ Y) (g : Y ⟶ Z),
{
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
Wal... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rintro _ _ _ (_|_|_) g | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left
Z✝ : WalkingParallelPair
g : one ⟶ Z✝
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.cases... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases g | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right
Z✝ : WalkingParallelPair
g : one ⟶ Z✝
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.case... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases g | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id
X✝ Z✝ : WalkingParallelPair
g : X✝ ⟶ Z✝
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.cases... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases g | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left.id
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.casesOn (motive := fun a a_1 t =>
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right.id
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.casesOn (motive := fun a a_1 t =>
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.left
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.casesOn (motive := fun a a_1 t =>
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.right
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.casesOn (motive := fun a a_1 t =>
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id.id
X✝ : WalkingParallelPair
⊢ {
obj := fun x =>
op
(WalkingParallelPair.casesOn (motive := fun t => x = t → WalkingParallelPair) x (fun h => one)
(fun h => zero) (_ : x = x)),
map := fun {X Y} f =>
WalkingParallelPairHom.casesOn (motive :... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x := op <| by cases x; exacts [one, zero]
map f := by
cases f <;> apply Quiver.Hom.op
exacts [left, right, WalkingParall... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.129_0.eJEUq2AFfmN187w | /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where
obj x | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
j : WalkingParallelPair
⊢ (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj j | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases j | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case zero
⊢ (𝟭 WalkingParallelPair).obj zero = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj zero | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case one
⊢ (𝟭 WalkingParallelPair).obj one = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj one | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
⊢ ∀ {X Y : WalkingParallelPair} (f : X ⟶ Y),
(𝟭 WalkingParallelPair).map f ≫
((fun j =>
eqToIso
(_ : (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj j))
Y).hom =
((fun j =>
eqToIso
(_ :... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rintro _ _ (_ | _ | _) | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case left
⊢ (𝟭 WalkingParallelPair).map left ≫
((fun j =>
eqToIso (_ : (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj j))
one).hom =
((fun j =>
eqToIso (_ : (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParallel... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case right
⊢ (𝟭 WalkingParallelPair).map right ≫
((fun j =>
eqToIso (_ : (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj j))
one).hom =
((fun j =>
eqToIso (_ : (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParall... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case id
X✝ : WalkingParallelPair
⊢ (𝟭 WalkingParallelPair).map (WalkingParallelPairHom.id X✝) ≫
((fun j =>
eqToIso (_ : (𝟭 WalkingParallelPair).obj j = (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).obj j))
X✝).hom =
((fun j =>
eqToIso (_ : (𝟭 WalkingParallelPair).... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
j : WalkingParallelPairᵒᵖ
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj j = (𝟭 WalkingParallelPairᵒᵖ).obj j | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | induction' j with X | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h
X : WalkingParallelPair
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj (op X) = (𝟭 WalkingParallelPairᵒᵖ).obj (op X) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | cases X | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h.zero
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj (op zero) = (𝟭 WalkingParallelPairᵒᵖ).obj (op zero) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h.one
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj (op one) = (𝟭 WalkingParallelPairᵒᵖ).obj (op one) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rfl | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
i j : WalkingParallelPairᵒᵖ
f : i ⟶ j
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).map f ≫
((fun j =>
eqToIso
(_ : (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj j = (𝟭 WalkingParallelPairᵒᵖ).obj j))
j).hom =
((fun j =>
eqToIso
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | induction' i with i | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h
j : WalkingParallelPairᵒᵖ
i : WalkingParallelPair
f : op i ⟶ j
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).map f ≫
((fun j =>
eqToIso
(_ : (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj j = (𝟭 WalkingParallelPairᵒᵖ).obj j))
j).hom =
((fun j ... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | induction' j with j | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h.h
i j : WalkingParallelPair
f : op i ⟶ op j
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).map f ≫
((fun j =>
eqToIso
(_ : (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj j = (𝟭 WalkingParallelPairᵒᵖ).obj j))
(op j)).hom =
((fun j =>
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | let g := f.unop | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h.h
i j : WalkingParallelPair
f : op i ⟶ op j
g : (op j).unop ⟶ (op i).unop := f.unop
⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).map f ≫
((fun j =>
eqToIso
(_ : (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).obj j = (𝟭 WalkingParallelPairᵒᵖ).obj j))
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | have : f = g.op := rfl | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor := walkingParallelPairOp
inverse := walkingParallelPairOp.leftOp
unitIso :=
NatIso.o... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.158_0.eJEUq2AFfmN187w | /--
The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to
right.
-/
@[simps functor inverse]
def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where
functor | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
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