state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case h.e'_3
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
⊢ T R ((m + 2) * n) + T R (m * n) = T R (2 * n + m * n) + T R (m * n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring_nf | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n... | Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
this : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)
⊢ T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [this, T_mul m, ← T_mul (m + 1)] | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n... | Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ Linear k (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | instance : Linear k (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.62_0.ADbOgJGW1JDvdmK | instance : Linear k (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ AddCommGroup (CoeSort.coe V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | instance (V : FdRep k G) : AddCommGroup V := by
| Mathlib.RepresentationTheory.FdRep.67_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : AddCommGroup V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ AddCommGroup ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; | Mathlib.RepresentationTheory.FdRep.67_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : AddCommGroup V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ Module k (CoeSort.coe V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | instance (V : FdRep k G) : Module k V := by
| Mathlib.RepresentationTheory.FdRep.70_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : Module k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ Module k ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; | Mathlib.RepresentationTheory.FdRep.70_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : Module k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ FiniteDimensional k (CoeSort.coe V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V) | instance (V : FdRep k G) : FiniteDimensional k V := by
| Mathlib.RepresentationTheory.FdRep.73_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : FiniteDimensional k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ FiniteDimensional k ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); | Mathlib.RepresentationTheory.FdRep.73_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : FiniteDimensional k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V W : FdRep k G
i : V ≅ W
g : G
⊢ (ρ W) g = (LinearEquiv.conj (isoToLinearEquiv i)) ((ρ V) g) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply] | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
| Mathlib.RepresentationTheory.FdRep.91_0.ADbOgJGW1JDvdmK | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V W : FdRep k G
i : V ≅ W
g : G
⊢ (ρ W) g =
((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv ≫
(ρ V) g ≫ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)] | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
| Mathlib.RepresentationTheory.FdRep.91_0.ADbOgJGW1JDvdmK | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V W : FdRep k G
i : V ≅ W
g : G
⊢ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom ≫ (ρ W) g =
(ρ V) g ≫ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | exact (i.hom.comm g).symm | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
| Mathlib.RepresentationTheory.FdRep.91_0.ADbOgJGW1JDvdmK | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ Rep.ρ ((forget₂ (FdRep k G) (Rep k G)).obj V) = ρ V | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | ext g v | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
| Mathlib.RepresentationTheory.FdRep.109_0.ADbOgJGW1JDvdmK | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ | Mathlib_RepresentationTheory_FdRep |
case h.h
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
g : G
v : CoeSort.coe ((forget₂ (FdRep k G) (Rep k G)).obj V)
⊢ ((Rep.ρ ((forget₂ (FdRep k G) (Rep k G)).obj V)) g) v = ((ρ V) g) v | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | rfl | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; | Mathlib.RepresentationTheory.FdRep.109_0.ADbOgJGW1JDvdmK | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ MonoidalCategory (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | example : MonoidalCategory (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.114_0.ADbOgJGW1JDvdmK | example : MonoidalCategory (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ MonoidalPreadditive (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | example : MonoidalPreadditive (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.116_0.ADbOgJGW1JDvdmK | example : MonoidalPreadditive (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ MonoidalLinear k (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | example : MonoidalLinear k (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.118_0.ADbOgJGW1JDvdmK | example : MonoidalLinear k (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ HasKernels (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | instance : HasKernels (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.126_0.ADbOgJGW1JDvdmK | instance : HasKernels (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝ : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y
⊢ (fun f => Action.Hom.mk ((forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom))
(AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | ext | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl... | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
case h.h
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝¹ : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y
x✝ : ↑((forget₂ (FdRep k G) (Rep k G)).obj X).V
⊢ ((fun f => Action.Hom.mk ((forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom))
(AddHom.toFun
... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | rfl | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl... | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝ : X ⟶ Y
⊢ AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | ext | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl... | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
case h.w
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝¹ : X ⟶ Y
x✝ : (forget (FGModuleCat k)).obj X.V
⊢ (AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_1 :... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | rfl | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl... | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Group G
⊢ RightRigidCategory (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)) | noncomputable instance : RightRigidCategory (FdRep k G) := by
| Mathlib.RepresentationTheory.FdRep.153_0.ADbOgJGW1JDvdmK | noncomputable instance : RightRigidCategory (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Group G
⊢ RightRigidCategory (Action (FGModuleCat k) ((forget₂ GroupCat MonCat).obj (GroupCat.of G))) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | infer_instance | noncomputable instance : RightRigidCategory (FdRep k G) := by
change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)); | Mathlib.RepresentationTheory.FdRep.153_0.ADbOgJGW1JDvdmK | noncomputable instance : RightRigidCategory (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G V : Type u
inst✝⁴ : Field k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : FiniteDimensional k V
ρV : Representation k G V
W : FdRep k G
⊢ of (dual ρV) ⊗ W ≅ of (linHom ρV (ρ W)) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | refine Action.mkIso (dualTensorIsoLinHomAux ρV W) ?_ | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) := by
| Mathlib.RepresentationTheory.FdRep.182_0.ADbOgJGW1JDvdmK | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) | Mathlib_RepresentationTheory_FdRep |
k G V : Type u
inst✝⁴ : Field k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : FiniteDimensional k V
ρV : Representation k G V
W : FdRep k G
⊢ ∀ (g : ↑(MonCat.of G)),
(of (dual ρV) ⊗ W).ρ g ≫ (dualTensorIsoLinHomAux ρV W).hom =
(dualTensorIsoLinHomAux ρV W).hom ≫ (of (linHom ρV (ρ W))).ρ... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.Representati... | convert dualTensorHom_comm ρV W.ρ | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) := by
refine Action.mkIso (dualTensorIsoLinHomAu... | Mathlib.RepresentationTheory.FdRep.182_0.ADbOgJGW1JDvdmK | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) | Mathlib_RepresentationTheory_FdRep |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : SupSet α
inst✝ : Inhabited ↑s
t : Set ↑s
h' : Set.Nonempty t
h'' : BddAbove t
h : sSup (Subtype.val '' t) ∈ s
⊢ sSup (Subtype.val '' t) = ↑(sSup t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [dif_pos, h, h', h''] | theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by | Mathlib.Order.CompleteLatticeIntervals.57_0.e28Rmw8JX0zQo3b | theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : SupSet α
inst✝ : Inhabited ↑s
⊢ sSup ∅ = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [sSup] | theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
| Mathlib.Order.CompleteLatticeIntervals.62_0.e28Rmw8JX0zQo3b | theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : SupSet α
inst✝ : Inhabited ↑s
t : Set ↑s
ht : ¬BddAbove t
⊢ sSup t = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [sSup, ht] | theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
| Mathlib.Order.CompleteLatticeIntervals.66_0.e28Rmw8JX0zQo3b | theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : InfSet α
inst✝ : Inhabited ↑s
t : Set ↑s
h' : Set.Nonempty t
h'' : BddBelow t
h : sInf (Subtype.val '' t) ∈ s
⊢ sInf (Subtype.val '' t) = ↑(sInf t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [dif_pos, h, h', h''] | theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by | Mathlib.Order.CompleteLatticeIntervals.97_0.e28Rmw8JX0zQo3b | theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : InfSet α
inst✝ : Inhabited ↑s
⊢ sInf ∅ = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [sInf] | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
| Mathlib.Order.CompleteLatticeIntervals.102_0.e28Rmw8JX0zQo3b | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : InfSet α
inst✝ : Inhabited ↑s
t : Set ↑s
ht : ¬BddBelow t
⊢ sInf t = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [sInf, ht] | theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
| Mathlib.Order.CompleteLatticeIntervals.106_0.e28Rmw8JX0zQo3b | theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rintro t c h_bdd hct | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe _).le_csSup_image hct h_bdd | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rintro t B ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).csSup_image_le ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rintro t c h_bdd hct | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).csInf_image_le hct h_bdd | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | intro t B ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).le_csInf_image ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [ht] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSe... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [ht] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable de... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
ht : Set.Nonempty t
h_bdd : BddAbove t
⊢ sSup (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
h_bdd : BddAbove t
c : ↑s
hct : c ∈ t
⊢ sSup (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ upperBounds t
⊢ sSup (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | refine' hs.out c.2 B.2 ⟨_, _⟩ | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_1
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ upperBounds t
⊢ ↑c ≤ sSup (Subtype.val '' t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩ | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_2
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ upperBounds t
⊢ sSup (Subtype.val '' t) ≤ ↑B | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
ht : Set.Nonempty t
h_bdd : BddBelow t
⊢ sInf (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
h_bdd : BddBelow t
c : ↑s
hct : c ∈ t
⊢ sInf (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ lowerBounds t
⊢ sInf (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | refine' hs.out B.2 c.2 ⟨_, _⟩ | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_1
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ lowerBounds t
⊢ ↑B ≤ sInf (Subtype.val '' t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_2
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ lowerBounds t
⊢ sInf (Subtype.val '' t) ≤ ↑c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩ | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t :... | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : ¬S = ∅
⊢ sSup (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ sSup (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ a ≤ sSup (Subtype.val '' S) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | obtain ⟨c, hc⟩ := hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ a ≤ sSup (Subtype.val '' S) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩) | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ c ≤ sSup S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ c ≤ sSup S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ c ≤ sSup S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ c ≤ { val := a, property := (_ : a ≤ a ∧ a ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [hS] at hc | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ c ≤ { val := sSup (Subtype.val '' S), property := (_ : a ≤ sSup (Subtype.val '' S) ∧ sSup (Subtype.val '' S) ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
⊢ sSup S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : S = ∅
⊢ sSup S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : ¬S = ∅
⊢ sSup S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : S = ∅
⊢ { val := a, property := (_ : a ≤ a ∧ a ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact c.2.1 | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : ¬S = ∅
⊢ { val := sSup (Subtype.val '' S), property := (_ : a ≤ sSup (Subtype.val '' S) ∧ sSup (Subtype.val '' S) ≤... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h) | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : ¬S = ∅
⊢ sInf (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ sInf (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ sInf (Subtype.val '' S) ≤ b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | obtain ⟨c, hc⟩ := hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ sInf (Subtype.val '' S) ≤ b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2 | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ sInf S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ sInf S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ sInf S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ { val := b, property := (_ : a ≤ b ∧ b ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp [hS] at hc | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ { val := sInf (Subtype.val '' S), property := (_ : a ≤ sInf (Subtype.val '' S) ∧ sInf (Subtype.val '' S) ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
⊢ c ≤ sInf S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : S = ∅
⊢ c ≤ sInf S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : ¬S = ∅
⊢ c ≤ sInf S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : S = ∅
⊢ c ≤ { val := b, property := (_ : a ≤ b ∧ b ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact c.2.2 | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : ¬S = ∅
⊢ c ≤ { val := sInf (Subtype.val '' S), property := (_ : a ≤ sInf (Subtype.val '' S) ∧ sInf (Subtype.val '' ... | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.co... | exact le_csInf ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h) | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_... | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ inner ((Equiv.pointReflection c) p₁ -ᵥ p₂) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc] | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
| Mathlib.Geometry.Euclidean.PerpBisector.54_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ 2⁻¹ * inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 ↔ inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
| Mathlib.Geometry.Euclidean.PerpBisector.54_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ ((Equiv.pointReflection p₂) p₁) ↔ inner (c -ᵥ p₂) (p₁ -ᵥ p₂) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev] | theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
| Mathlib.Geometry.Euclidean.PerpBisector.60_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp [mem_perpBisector_iff_inner_eq_zero] | theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.66_0.WKtplj3xHYGfYbJ | theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ direction (perpBisector p₁ p₂) = (Submodule.span ℝ {p₂ -ᵥ p₁})ᗮ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction] | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
| Mathlib.Geometry.Euclidean.PerpBisector.73_0.WKtplj3xHYGfYbJ | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ LinearMap.ker ((innerₛₗ ℝ) (p₂ -ᵥ p₁)) = (Submodule.span ℝ {p₂ -ᵥ p₁})ᗮ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | ext x | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
| Mathlib.Geometry.Euclidean.PerpBisector.73_0.WKtplj3xHYGfYbJ | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ | Mathlib_Geometry_Euclidean_PerpBisector |
case h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
x : V
⊢ x ∈ LinearMap.ker ((innerₛₗ ℝ) (p₂ -ᵥ p₁)) ↔ x ∈ (Submodule.span ℝ {p₂ -ᵥ p₁})ᗮ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
| Mathlib.Geometry.Euclidean.PerpBisector.73_0.WKtplj3xHYGfYbJ | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ inner (c -ᵥ p₁) (p₂ -ᵥ p₁) = inner (c -ᵥ p₂) (p₁ -ᵥ p₂) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left] | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
| Mathlib.Geometry.Euclidean.PerpBisector.81_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 ↔ 2⁻¹ * inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_le... | Mathlib.Geometry.Euclidean.PerpBisector.81_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ inner (c -ᵥ p₁) (p₂ -ᵥ p₁) = dist p₁ p₂ ^ 2 / 2 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul] | theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
| Mathlib.Geometry.Euclidean.PerpBisector.87_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner] | theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.93_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp only [mem_perpBisector_iff_dist_eq, dist_comm] | theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
| Mathlib.Geometry.Euclidean.PerpBisector.98_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ perpBisector p₁ p₂ = perpBisector p₂ p₁ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | ext c | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
| Mathlib.Geometry.Euclidean.PerpBisector.101_0.WKtplj3xHYGfYbJ | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ | Mathlib_Geometry_Euclidean_PerpBisector |
case h
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c✝ c₁ c₂ p₁✝ p₂✝ p₁ p₂ c : P
⊢ c ∈ perpBisector p₁ p₂ ↔ c ∈ perpBisector p₂ p₁ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp only [mem_perpBisector_iff_dist_eq, eq_comm] | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
ext c; | Mathlib.Geometry.Euclidean.PerpBisector.101_0.WKtplj3xHYGfYbJ | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simpa [mem_perpBisector_iff_inner_eq_inner] using eq_comm | @[simp] theorem right_mem_perpBisector : p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.104_0.WKtplj3xHYGfYbJ | @[simp] theorem right_mem_perpBisector : p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [perpBisector_comm, right_mem_perpBisector, eq_comm] | @[simp] theorem left_mem_perpBisector : p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.107_0.WKtplj3xHYGfYbJ | @[simp] theorem left_mem_perpBisector : p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
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