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 tail.head.head
n : ℕ
h : 2 ≤ n
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simp [pathGraph, ← Fin.coe_covby_iff] | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
fin_cases... | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case tail.head.tail.head
n : ℕ
h : 2 ≤ n
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simp [pathGraph, ← Fin.coe_covby_iff] | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
fin_cases... | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
⊢ chromaticNumber (pathGraph n) = 2 | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | have hc := (pathGraph.bicoloring n).to_colorable | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 := by
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.43_0.jXeFS7nTQciTQGN | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
hc : Colorable (pathGraph n) (Fintype.card Bool)
⊢ chromaticNumber (pathGraph n) = 2 | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | apply le_antisymm | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 := by
have hc := (pathGraph.bicoloring n).to_colorable
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.43_0.jXeFS7nTQciTQGN | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case a
n : ℕ
h : 2 ≤ n
hc : Colorable (pathGraph n) (Fintype.card Bool)
⊢ chromaticNumber (pathGraph n) ≤ 2 | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | exact chromaticNumber_le_of_colorable hc | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 := by
have hc := (pathGraph.bicoloring n).to_colorable
apply le_antisymm
· | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.43_0.jXeFS7nTQciTQGN | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case a
n : ℕ
h : 2 ≤ n
hc : Colorable (pathGraph n) (Fintype.card Bool)
⊢ 2 ≤ chromaticNumber (pathGraph n) | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simpa only [pathGraph_two_eq_top, chromaticNumber_top] using
hc.chromaticNumber_mono_of_embedding (pathGraph_two_embedding n h) | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 := by
have hc := (pathGraph.bicoloring n).to_colorable
apply le_antisymm
· exact chromaticNumber_le_of_colorable hc
· | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.43_0.jXeFS7nTQciTQGN | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed K
f : k →+* K
p : k[X]
h : Separable p
⊢ Splits f p | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | convert IsSepClosed.splits_of_separable (p.map f) (Separable.map h) | /-- Every separable polynomial splits in the field extension `f : k →+* K` if `K` is
separably closed.
See also `IsSepClosed.splits_domain` for the case where `k` is separably closed.
-/
theorem IsSepClosed.splits_codomain [IsSepClosed K] {f : k →+* K}
(p : k[X]) (h : p.Separable) : p.Splits f := by
| Mathlib.FieldTheory.IsSepClosed.70_0.3ZRqn1f8ZTqE2nc | /-- Every separable polynomial splits in the field extension `f : k →+* K` if `K` is
separably closed.
See also `IsSepClosed.splits_domain` for the case where `k` is separably closed.
-/
theorem IsSepClosed.splits_codomain [IsSepClosed K] {f : k →+* K}
(p : k[X]) (h : p.Separable) : p.Splits f | Mathlib_FieldTheory_IsSepClosed |
case a
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed K
f : k →+* K
p : k[X]
h : Separable p
⊢ Splits f p ↔ Splits (RingHom.id K) (map f p) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | simp [splits_map_iff] | /-- Every separable polynomial splits in the field extension `f : k →+* K` if `K` is
separably closed.
See also `IsSepClosed.splits_domain` for the case where `k` is separably closed.
-/
theorem IsSepClosed.splits_codomain [IsSepClosed K] {f : k →+* K}
(p : k[X]) (h : p.Separable) : p.Splits f := by
convert IsSe... | Mathlib.FieldTheory.IsSepClosed.70_0.3ZRqn1f8ZTqE2nc | /-- Every separable polynomial splits in the field extension `f : k →+* K` if `K` is
separably closed.
See also `IsSepClosed.splits_domain` for the case where `k` is separably closed.
-/
theorem IsSepClosed.splits_codomain [IsSepClosed K] {f : k →+* K}
(p : k[X]) (h : p.Separable) : p.Splits f | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
⊢ ∃ z, z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
| Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
h : n = 0
⊢ False | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [h, Nat.cast_zero] at hn | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
| Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero 0
h : n = 0
⊢ False | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact hn.out rfl | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
| Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
⊢ ∃ z, z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.2 hn').ne' | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
| Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
⊢ degree (X ^ n - C x) ≠ 0 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [degree_X_pow_sub_C hn' x] | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
| Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
⊢ ↑n ≠ 0 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact (WithBot.coe_lt_coe.2 hn').ne' | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
| Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
this : degree (X ^ n - C x) ≠ 0
⊢ ∃ z, z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | by_cases hx : x = 0 | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.... | Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
case pos
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
this : degree (X ^ n - C x) ≠ 0
hx : x = 0
⊢ ∃ z, z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact ⟨0, by rw [hx, pow_eq_zero_iff hn']⟩ | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.... | Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
this : degree (X ^ n - C x) ≠ 0
hx : x = 0
⊢ 0 ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [hx, pow_eq_zero_iff hn'] | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.... | Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
case neg
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
this : degree (X ^ n - C x) ≠ 0
hx : ¬x = 0
⊢ ∃ z, z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | obtain ⟨z, hz⟩ := exists_root _ this <| separable_X_pow_sub_C x hn.out hx | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.... | Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
case neg.intro
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
this : degree (X ^ n - C x) ≠ 0
hx : ¬x = 0
z : k
hz : IsRoot (X ^ n - C x) z
⊢ ∃ z, z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | use z | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.... | Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
case h
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
n : ℕ
hn : NeZero ↑n
hn' : 0 < n
this : degree (X ^ n - C x) ≠ 0
hx : ¬x = 0
z : k
hz : IsRoot (X ^ n - C x) z
⊢ z ^ n = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | simpa [eval_C, eval_X, eval_pow, eval_sub, IsRoot.def, sub_eq_zero] using hz | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero <| fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.... | Mathlib.FieldTheory.IsSepClosed.94_0.3ZRqn1f8ZTqE2nc | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
x : k
h2 : NeZero 2
⊢ ∃ z, x = z * z | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rcases exists_pow_nat_eq x 2 with ⟨z, rfl⟩ | theorem exists_eq_mul_self [IsSepClosed k] (x : k) [h2 : NeZero (2 : k)] : ∃ z, x = z * z := by
| Mathlib.FieldTheory.IsSepClosed.108_0.3ZRqn1f8ZTqE2nc | theorem exists_eq_mul_self [IsSepClosed k] (x : k) [h2 : NeZero (2 : k)] : ∃ z, x = z * z | Mathlib_FieldTheory_IsSepClosed |
case intro
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
h2 : NeZero 2
z : k
⊢ ∃ z_1, z ^ 2 = z_1 * z_1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact ⟨z, sq z⟩ | theorem exists_eq_mul_self [IsSepClosed k] (x : k) [h2 : NeZero (2 : k)] : ∃ z, x = z * z := by
rcases exists_pow_nat_eq x 2 with ⟨z, rfl⟩
| Mathlib.FieldTheory.IsSepClosed.108_0.3ZRqn1f8ZTqE2nc | theorem exists_eq_mul_self [IsSepClosed k] (x : k) [h2 : NeZero (2 : k)] : ∃ z, x = z * z | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
⊢ roots p = 0 ↔ p = C (coeff p 0) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | refine' ⟨fun h => _, fun hp => by rw [hp, roots_C]⟩ | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
| Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
hp : p = C (coeff p 0)
⊢ roots p = 0 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [hp, roots_C] | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
refine' ⟨fun h => _, fun hp => by | Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
h : roots p = 0
⊢ p = C (coeff p 0) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rcases le_or_lt (degree p) 0 with hd | hd | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
refine' ⟨fun h => _, fun hp => by rw [hp, roots_C]⟩
| Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
case inl
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
h : roots p = 0
hd : degree p ≤ 0
⊢ p = C (coeff p 0) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact eq_C_of_degree_le_zero hd | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
refine' ⟨fun h => _, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· | Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
case inr
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
h : roots p = 0
hd : 0 < degree p
⊢ p = C (coeff p 0) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | obtain ⟨z, hz⟩ := IsSepClosed.exists_root p hd.ne' hsep | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
refine' ⟨fun h => _, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· exact eq_C_of_degree_le_zero hd
· | Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
case inr.intro
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
h : roots p = 0
hd : 0 < degree p
z : k
hz : IsRoot p z
⊢ p = C (coeff p 0) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [← mem_roots (ne_zero_of_degree_gt hd), h] at hz | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
refine' ⟨fun h => _, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· exact eq_C_of_degree_le_zero hd
· obtain ⟨z, hz⟩ := IsSepClosed.exists_root p hd.ne' hs... | Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
case inr.intro
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
p : k[X]
hsep : Separable p
h : roots p = 0
hd : 0 < degree p
z : k
hz : z ∈ 0
⊢ p = C (coeff p 0) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | simp at hz | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by
refine' ⟨fun h => _, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· exact eq_C_of_degree_le_zero hd
· obtain ⟨z, hz⟩ := IsSepClosed.exists_root p hd.ne' hs... | Mathlib.FieldTheory.IsSepClosed.112_0.3ZRqn1f8ZTqE2nc | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed K
f : k →+* K
p : k[X]
hp : degree p ≠ 0
hsep : Separable p
⊢ degree (map f p) ≠ 0 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rwa [degree_map_eq_of_injective f.injective] | theorem exists_eval₂_eq_zero [IsSepClosed K] (f : k →+* K)
(p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable) :
∃ x, p.eval₂ f x = 0 :=
let ⟨x, hx⟩ := exists_root (p.map f) (by | Mathlib.FieldTheory.IsSepClosed.121_0.3ZRqn1f8ZTqE2nc | theorem exists_eval₂_eq_zero [IsSepClosed K] (f : k →+* K)
(p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable) :
∃ x, p.eval₂ f x = 0 | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed K
f : k →+* K
p : k[X]
hp : degree p ≠ 0
hsep : Separable p
x : K
hx : IsRoot (map f p) x
⊢ eval₂ f x p = 0 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rwa [eval₂_eq_eval_map, ← IsRoot] | theorem exists_eval₂_eq_zero [IsSepClosed K] (f : k →+* K)
(p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable) :
∃ x, p.eval₂ f x = 0 :=
let ⟨x, hx⟩ := exists_root (p.map f) (by rwa [degree_map_eq_of_injective f.injective])
(Separable.map hsep)
⟨x, by | Mathlib.FieldTheory.IsSepClosed.121_0.3ZRqn1f8ZTqE2nc | theorem exists_eval₂_eq_zero [IsSepClosed K] (f : k →+* K)
(p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable) :
∃ x, p.eval₂ f x = 0 | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
⊢ IsSepClosed k | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | refine ⟨fun p hsep ↦ Or.inr ?_⟩ | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
| Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
⊢ ∀ {g : k[X]}, Irreducible g → g ∣ map (RingHom.id k) p → degree g = 1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | intro q hq hdvd | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
| Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ map (RingHom.id k) p
⊢ degree q = 1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | simp only [map_id] at hdvd | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
| Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
⊢ degree q = 1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_zero hq | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
| Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
hlc : IsUnit (leadingCoeff q)⁻¹
⊢ degree q = 1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have hsep' : Separable (q * C (leadingCoeff q)⁻¹) :=
Separable.mul (Separable.of_dvd hsep hdvd) ((separable_C _).2 hlc)
(by simpa only [← isCoprime_mul_unit_right_right (isUnit_C.2 hlc) q 1, one_mul]
using isCoprime_one_right (x := q)) | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_... | Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
hlc : IsUnit (leadingCoeff q)⁻¹
⊢ IsCoprime q (C (leadingCoeff q)⁻¹) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | simpa only [← isCoprime_mul_unit_right_right (isUnit_C.2 hlc) q 1, one_mul]
using isCoprime_one_right (x := q) | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_... | Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
hlc : IsUnit (leadingCoeff q)⁻¹
hsep' : Separable (q * C (leadingCoeff q)⁻¹)
⊢ degree q = 1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have hirr' := hq | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_... | Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
hlc : IsUnit (leadingCoeff q)⁻¹
hsep' : Separable (q * C (leadingCoeff q)⁻¹)
hirr' : Irreducible q
⊢ degree q = 1 | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [← irreducible_mul_isUnit (isUnit_C.2 hlc)] at hirr' | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_... | Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
hlc : IsUnit (leadingCoeff q)⁻¹
hsep' : Separable (q * C (leadingCoeff q)⁻¹)
hirr' : Irreducible (q * C (leadingCoeff q)... | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | obtain ⟨x, hx⟩ := H (q * C (leadingCoeff q)⁻¹) (monic_mul_leadingCoeff_inv hq.ne_zero) hirr' hsep' | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_... | Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
case intro
k : Type u
inst✝¹ : Field k
K : Type v
inst✝ : Field K
H : ∀ (p : k[X]), Monic p → Irreducible p → Separable p → ∃ x, eval x p = 0
p : k[X]
hsep : Separable p
q : k[X]
hq : Irreducible q
hdvd : q ∣ p
hlc : IsUnit (leadingCoeff q)⁻¹
hsep' : Separable (q * C (leadingCoeff q)⁻¹)
hirr' : Irreducible (q * C (lead... | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact degree_mul_leadingCoeff_inv q hq.ne_zero ▸ degree_eq_one_of_irreducible_of_root hirr' hx | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k := by
refine ⟨fun p hsep ↦ Or.inr ?_⟩
intro q hq hdvd
simp only [map_id] at hdvd
have hlc : IsUnit (leadingCoeff q)⁻¹ := IsUnit.inv <| Ne.isUnit <|
leadingCoeff_ne_zero.2 <| Irreducible.ne_... | Mathlib.FieldTheory.IsSepClosed.136_0.3ZRqn1f8ZTqE2nc | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
IsSepClosed k | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
⊢ Function.Surjective ⇑(algebraMap k K) | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩ | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
| Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
x : K
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x) | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
| Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
x : K
hq : leadingCoeff (minpoly k x) = 1
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have hsep : (minpoly k x).Separable := IsSeparable.separable k x | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
| Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
x : K
hq : leadingCoeff (minpoly k x) = 1
hsep : Separable (minpoly k x)
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have h : (minpoly k x).degree = 1 :=
degree_eq_one_of_irreducible k (minpoly.irreducible (IsSeparable.isIntegral k x)) hsep | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
have hsep : (minpoly k x).Separable := IsSeparable.... | Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
x : K
hq : leadingCoeff (minpoly k x) = 1
hsep : Separable (minpoly k x)
h : degree (minpoly k x) = 1
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | have : aeval x (minpoly k x) = 0 := minpoly.aeval k x | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
have hsep : (minpoly k x).Separable := IsSeparable.... | Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
x : K
hq : leadingCoeff (minpoly k x) = 1
hsep : Separable (minpoly k x)
h : degree (minpoly k x) = 1
this : (aeval x) (minpoly k x) = 0
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | rw [eq_X_add_C_of_degree_eq_one h, hq, C_1, one_mul, aeval_add, aeval_X, aeval_C,
add_eq_zero_iff_eq_neg] at this | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
have hsep : (minpoly k x).Separable := IsSeparable.... | Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : IsSeparable k K
x : K
hq : leadingCoeff (minpoly k x) = 1
hsep : Separable (minpoly k x)
h : degree (minpoly k x) = 1
this : x = -(algebraMap k K) (coeff (minpoly k x) 0)
⊢ (algebraMap k K) (-coeff (minpoly k x) ... | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | exact (RingHom.map_neg (algebraMap k K) ((minpoly k x).coeff 0)).symm ▸ this.symm | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
have hsep : (minpoly k x).Separable := IsSeparable.... | Mathlib.FieldTheory.IsSepClosed.158_0.3ZRqn1f8ZTqE2nc | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsSepClosed |
k : Type u
inst✝² : Field k
K : Type v
inst✝¹ : Field K
inst✝ : IsSepClosed k
⊢ IsSepClosed k | /-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and pr... | assumption | /-- A separably closed field is its separable closure. -/
instance IsSepClosure.self_of_isSepClosed [IsSepClosed k] : IsSepClosure k k :=
⟨by | Mathlib.FieldTheory.IsSepClosed.180_0.3ZRqn1f8ZTqE2nc | /-- A separably closed field is its separable closure. -/
instance IsSepClosure.self_of_isSepClosed [IsSepClosed k] : IsSepClosure k k | Mathlib_FieldTheory_IsSepClosed |
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ CompactSpace ↑(characterSpace 𝕜 A) | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.WeakDual
import Mathlib.Analysis.NormedSpace.Spectrum
#align_import analy... | rw [← isCompact_iff_compactSpace] | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
| Mathlib.Analysis.NormedSpace.Algebra.45_0.qrhY2kMk1fwIMuY | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) | Mathlib_Analysis_NormedSpace_Algebra |
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ IsCompact (characterSpace 𝕜 A) | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.WeakDual
import Mathlib.Analysis.NormedSpace.Spectrum
#align_import analy... | have h : characterSpace 𝕜 A ⊆ toNormedDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
intro φ hφ
rw [Set.mem_preimage, mem_closedBall_zero_iff]
exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ : _) | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
rw [← isCompact_iff_compactSpace]
| Mathlib.Analysis.NormedSpace.Algebra.45_0.qrhY2kMk1fwIMuY | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) | Mathlib_Analysis_NormedSpace_Algebra |
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ characterSpace 𝕜 A ⊆ ⇑toNormedDual ⁻¹' Metric.closedBall 0 ‖1‖ | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.WeakDual
import Mathlib.Analysis.NormedSpace.Spectrum
#align_import analy... | intro φ hφ | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
rw [← isCompact_iff_compactSpace]
have h : characterSpace 𝕜 A ⊆ toNormedDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
| Mathlib.Analysis.NormedSpace.Algebra.45_0.qrhY2kMk1fwIMuY | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) | Mathlib_Analysis_NormedSpace_Algebra |
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ φ ∈ ⇑toNormedDual ⁻¹' Metric.closedBall 0 ‖1‖ | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.WeakDual
import Mathlib.Analysis.NormedSpace.Spectrum
#align_import analy... | rw [Set.mem_preimage, mem_closedBall_zero_iff] | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
rw [← isCompact_iff_compactSpace]
have h : characterSpace 𝕜 A ⊆ toNormedDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
intro φ hφ
| Mathlib.Analysis.NormedSpace.Algebra.45_0.qrhY2kMk1fwIMuY | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) | Mathlib_Analysis_NormedSpace_Algebra |
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ ‖toNormedDual φ‖ ≤ ‖1‖ | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.WeakDual
import Mathlib.Analysis.NormedSpace.Spectrum
#align_import analy... | exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ : _) | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
rw [← isCompact_iff_compactSpace]
have h : characterSpace 𝕜 A ⊆ toNormedDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
intro φ hφ
rw [Set.mem_preimage, mem_closedBall_zero_iff]
| Mathlib.Analysis.NormedSpace.Algebra.45_0.qrhY2kMk1fwIMuY | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) | Mathlib_Analysis_NormedSpace_Algebra |
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
h : characterSpace 𝕜 A ⊆ ⇑toNormedDual ⁻¹' Metric.closedBall 0 ‖1‖
⊢ IsCompact (characterSpace 𝕜 A) | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.WeakDual
import Mathlib.Analysis.NormedSpace.Spectrum
#align_import analy... | exact (isCompact_closedBall 𝕜 0 _).of_isClosed_subset CharacterSpace.isClosed h | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
rw [← isCompact_iff_compactSpace]
have h : characterSpace 𝕜 A ⊆ toNormedDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
intro φ hφ
rw [Set.mem_preimage, mem_closedBall_zero_iff]
exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ : _)
| Mathlib.Analysis.NormedSpace.Algebra.45_0.qrhY2kMk1fwIMuY | instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) | Mathlib_Analysis_NormedSpace_Algebra |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g✝ : G → E
f'✝ g'✝ : G →L[ℝ] E
s✝ : Set G
x... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simpa using (hf.hasFDerivWithinAt.inner 𝕜 hg.hasFDerivWithinAt).hasDerivWithinAt | theorem HasDerivWithinAt.inner {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
HasDerivWithinAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) s x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.111_0.6FECEGgqdb67QLM | theorem HasDerivWithinAt.inner {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
HasDerivWithinAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) s x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g✝ : G → E
f'✝ g'✝ : G →L[ℝ] E
s : Set G
x✝... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simpa only [← hasDerivWithinAt_univ] using HasDerivWithinAt.inner 𝕜 | theorem HasDerivAt.inner {f g : ℝ → E} {f' g' : E} {x : ℝ} :
HasDerivAt f f' x → HasDerivAt g g' x →
HasDerivAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x :=
by | Mathlib.Analysis.InnerProductSpace.Calculus.117_0.6FECEGgqdb67QLM | theorem HasDerivAt.inner {f g : ℝ → E} {f' g' : E} {x : ℝ} :
HasDerivAt f f' x → HasDerivAt g g' x →
HasDerivAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [(hf.hasFDerivAt.inner 𝕜 hg.hasFDerivAt).fderiv] | theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ := by
| Mathlib.Analysis.InnerProductSpace.Calculus.142_0.6FECEGgqdb67QLM | theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ := by
rw [(hf.hasFDerivAt.inner 𝕜 hg.hasFDerivAt).fderiv]; | Mathlib.Analysis.InnerProductSpace.Calculus.142_0.6FECEGgqdb67QLM | theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | convert (reClm : 𝕜 →L[ℝ] ℝ).contDiff.comp ((contDiff_id (E := E)).inner 𝕜 (contDiff_id (E := E))) | theorem contDiff_norm_sq : ContDiff ℝ n fun x : E => ‖x‖ ^ 2 := by
| Mathlib.Analysis.InnerProductSpace.Calculus.153_0.6FECEGgqdb67QLM | theorem contDiff_norm_sq : ContDiff ℝ n fun x : E => ‖x‖ ^ 2 | Mathlib_Analysis_InnerProductSpace_Calculus |
case h.e'_10.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (inner_self_eq_norm_sq _).symm | theorem contDiff_norm_sq : ContDiff ℝ n fun x : E => ‖x‖ ^ 2 := by
convert (reClm : 𝕜 →L[ℝ] ℝ).contDiff.comp ((contDiff_id (E := E)).inner 𝕜 (contDiff_id (E := E)))
| Mathlib.Analysis.InnerProductSpace.Calculus.153_0.6FECEGgqdb67QLM | theorem contDiff_norm_sq : ContDiff ℝ n fun x : E => ‖x‖ ^ 2 | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x✝ : G... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | have : ‖id x‖ ^ 2 ≠ 0 := pow_ne_zero 2 (norm_pos_iff.2 hx).ne' | theorem contDiffAt_norm {x : E} (hx : x ≠ 0) : ContDiffAt ℝ n norm x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.171_0.6FECEGgqdb67QLM | theorem contDiffAt_norm {x : E} (hx : x ≠ 0) : ContDiffAt ℝ n norm x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x✝ : G... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simpa only [id, sqrt_sq, norm_nonneg] using (contDiffAt_id.norm_sq 𝕜).sqrt this | theorem contDiffAt_norm {x : E} (hx : x ≠ 0) : ContDiffAt ℝ n norm x := by
have : ‖id x‖ ^ 2 ≠ 0 := pow_ne_zero 2 (norm_pos_iff.2 hx).ne'
| Mathlib.Analysis.InnerProductSpace.Calculus.171_0.6FECEGgqdb67QLM | theorem contDiffAt_norm {x : E} (hx : x ≠ 0) : ContDiffAt ℝ n norm x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simp only [dist_eq_norm] | theorem ContDiffAt.dist (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x) :
ContDiffAt ℝ n (fun y => dist (f y) (g y)) x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.181_0.6FECEGgqdb67QLM | theorem ContDiffAt.dist (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x) :
ContDiffAt ℝ n (fun y => dist (f y) (g y)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) | theorem ContDiffAt.dist (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x) :
ContDiffAt ℝ n (fun y => dist (f y) (g y)) x := by
simp only [dist_eq_norm]
| Mathlib.Analysis.InnerProductSpace.Calculus.181_0.6FECEGgqdb67QLM | theorem ContDiffAt.dist (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x) :
ContDiffAt ℝ n (fun y => dist (f y) (g y)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simp only [dist_eq_norm] | theorem ContDiffWithinAt.dist (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
(hne : f x ≠ g x) : ContDiffWithinAt ℝ n (fun y => dist (f y) (g y)) s x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.192_0.6FECEGgqdb67QLM | theorem ContDiffWithinAt.dist (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
(hne : f x ≠ g x) : ContDiffWithinAt ℝ n (fun y => dist (f y) (g y)) s x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) | theorem ContDiffWithinAt.dist (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
(hne : f x ≠ g x) : ContDiffWithinAt ℝ n (fun y => dist (f y) (g y)) s x := by
simp only [dist_eq_norm]; | Mathlib.Analysis.InnerProductSpace.Calculus.192_0.6FECEGgqdb67QLM | theorem ContDiffWithinAt.dist (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
(hne : f x ≠ g x) : ContDiffWithinAt ℝ n (fun y => dist (f y) (g y)) s x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x✝ : G... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ] | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.220_0.6FECEGgqdb67QLM | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x✝ : G... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | convert (hasStrictFDerivAt_id x).inner ℝ (hasStrictFDerivAt_id x) | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x := by
simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ]
| Mathlib.Analysis.InnerProductSpace.Calculus.220_0.6FECEGgqdb67QLM | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
case h.e'_10.h.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | ext y | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x := by
simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ]
convert (hasStrictFDerivAt_id x).inner ℝ (hasStrictFDerivAt_id x)
| Mathlib.Analysis.InnerProductSpace.Calculus.220_0.6FECEGgqdb67QLM | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
case h.e'_10.h.h.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ]... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simp [two_smul, real_inner_comm] | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x := by
simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ]
convert (hasStrictFDerivAt_id x).inner ℝ (hasStrictFDerivAt_id x)
ext y
| Mathlib.Analysis.InnerProductSpace.Calculus.220_0.6FECEGgqdb67QLM | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g : G → E
f'✝ g' : G →L[ℝ] E
s : Set G
x✝ :... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simpa using hf.hasFDerivAt.norm_sq.hasDerivAt | theorem HasDerivAt.norm_sq {f : ℝ → F} {f' : F} {x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.232_0.6FECEGgqdb67QLM | theorem HasDerivAt.norm_sq {f : ℝ → F} {f' : F} {x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g : G → E
f'✝ g' : G →L[ℝ] E
s✝ : Set G
x✝ ... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simpa using hf.hasFDerivWithinAt.norm_sq.hasDerivWithinAt | theorem HasDerivWithinAt.norm_sq {f : ℝ → F} {f' : F} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') s x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.240_0.6FECEGgqdb67QLM | theorem HasDerivWithinAt.norm_sq {f : ℝ → F} {f' : F} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') s x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simp only [dist_eq_norm] | theorem DifferentiableAt.dist (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
(hne : f x ≠ g x) : DifferentiableAt ℝ (fun y => dist (f y) (g y)) x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.255_0.6FECEGgqdb67QLM | theorem DifferentiableAt.dist (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
(hne : f x ≠ g x) : DifferentiableAt ℝ (fun y => dist (f y) (g y)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) | theorem DifferentiableAt.dist (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
(hne : f x ≠ g x) : DifferentiableAt ℝ (fun y => dist (f y) (g y)) x := by
simp only [dist_eq_norm]; | Mathlib.Analysis.InnerProductSpace.Calculus.255_0.6FECEGgqdb67QLM | theorem DifferentiableAt.dist (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
(hne : f x ≠ g x) : DifferentiableAt ℝ (fun y => dist (f y) (g y)) x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | simp only [dist_eq_norm] | theorem DifferentiableWithinAt.dist (hf : DifferentiableWithinAt ℝ f s x)
(hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x) :
DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x := by
| Mathlib.Analysis.InnerProductSpace.Calculus.283_0.6FECEGgqdb67QLM | theorem DifferentiableWithinAt.dist (hf : DifferentiableWithinAt ℝ f s x)
(hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x) :
DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
f' g' : G →L[ℝ] E
s : Set G
x : G
... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) | theorem DifferentiableWithinAt.dist (hf : DifferentiableWithinAt ℝ f s x)
(hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x) :
DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x := by
simp only [dist_eq_norm]
| Mathlib.Analysis.InnerProductSpace.Calculus.283_0.6FECEGgqdb67QLM | theorem DifferentiableWithinAt.dist (hf : DifferentiableWithinAt ℝ f s x)
(hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x) :
DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ DifferentiableWithinAt 𝕜 f t y ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableWithinAt_iff, differentiableWithinAt_pi] | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y := by
| Mathlib.Analysis.InnerProductSpace.Calculus.312_0.6FECEGgqdb67QLM | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ (∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t y) ↔
∀ (i : ι... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableWithinAt_iff, differentiableWithinAt_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.312_0.6FECEGgqdb67QLM | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ DifferentiableAt 𝕜 f y ↔ ∀ (i : ι), DifferentiableAt 𝕜 (fun x => f x i) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableAt_iff, differentiableAt_pi] | theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y := by
| Mathlib.Analysis.InnerProductSpace.Calculus.318_0.6FECEGgqdb67QLM | theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ (∀ (i : ι), DifferentiableAt 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔
∀ (i : ι), Diffe... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableAt_iff, differentiableAt_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.318_0.6FECEGgqdb67QLM | theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ DifferentiableOn 𝕜 f t ↔ ∀ (i : ι), DifferentiableOn 𝕜 (fun x => f x i) t | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableOn_iff, differentiableOn_pi] | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t := by
| Mathlib.Analysis.InnerProductSpace.Calculus.324_0.6FECEGgqdb67QLM | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ (∀ (i : ι), DifferentiableOn 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t) ↔
∀ (i : ι), Diffe... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableOn_iff, differentiableOn_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.324_0.6FECEGgqdb67QLM | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ Differentiable 𝕜 f ↔ ∀ (i : ι), Differentiable 𝕜 fun x => f x i | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiable_iff, differentiable_pi] | theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i := by
| Mathlib.Analysis.InnerProductSpace.Calculus.330_0.6FECEGgqdb67QLM | theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ (∀ (i : ι), Differentiable 𝕜 fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) ↔
∀ (i : ι), Differentia... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiable_iff, differentiable_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.330_0.6FECEGgqdb67QLM | theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ HasStrictFDerivAt f f' y ↔ ∀ (i : ι), HasStrictFDerivAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') ... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi'] | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y := by
| Mathlib.Analysis.InnerProductSpace.Calculus.335_0.6FECEGgqdb67QLM | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ (∀ (i : ι),
HasStrictFDerivAt (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i)
(comp (proj i... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']
| Mathlib.Analysis.InnerProductSpace.Calculus.335_0.6FECEGgqdb67QLM | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ HasFDerivWithinAt f f' t y ↔ ∀ (i : ι), HasFDerivWithinAt (fun x => f x i) (comp (EuclideanSpace.proj i) f'... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi'] | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y := by
| Mathlib.Analysis.InnerProductSpace.Calculus.342_0.6FECEGgqdb67QLM | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
⊢ (∀ (i : ι),
HasFDerivWithinAt (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i)
(comp (proj i... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
| Mathlib.Analysis.InnerProductSpace.Calculus.342_0.6FECEGgqdb67QLM | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ ContDiffWithinAt 𝕜 n f t y ↔ ∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x => f x i) t y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffWithinAt_iff, contDiffWithinAt_pi] | theorem contDiffWithinAt_euclidean {n : ℕ∞} :
ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y := by
| Mathlib.Analysis.InnerProductSpace.Calculus.349_0.6FECEGgqdb67QLM | theorem contDiffWithinAt_euclidean {n : ℕ∞} :
ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ (∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t y) ↔
∀ (i ... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem contDiffWithinAt_euclidean {n : ℕ∞} :
ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffWithinAt_iff, contDiffWithinAt_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.349_0.6FECEGgqdb67QLM | theorem contDiffWithinAt_euclidean {n : ℕ∞} :
ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ ContDiffAt 𝕜 n f y ↔ ∀ (i : ι), ContDiffAt 𝕜 n (fun x => f x i) y | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffAt_iff, contDiffAt_pi] | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y := by
| Mathlib.Analysis.InnerProductSpace.Calculus.355_0.6FECEGgqdb67QLM | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ (∀ (i : ι), ContDiffAt 𝕜 n (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔
∀ (i : ι), Co... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffAt_iff, contDiffAt_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.355_0.6FECEGgqdb67QLM | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ ContDiffOn 𝕜 n f t ↔ ∀ (i : ι), ContDiffOn 𝕜 n (fun x => f x i) t | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffOn_iff, contDiffOn_pi] | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t := by
| Mathlib.Analysis.InnerProductSpace.Calculus.361_0.6FECEGgqdb67QLM | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ (∀ (i : ι), ContDiffOn 𝕜 n (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t) ↔
∀ (i : ι), Co... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffOn_iff, contDiffOn_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.361_0.6FECEGgqdb67QLM | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ ContDiff 𝕜 n f ↔ ∀ (i : ι), ContDiff 𝕜 n fun x => f x i | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiff_iff, contDiff_pi] | theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i := by
| Mathlib.Analysis.InnerProductSpace.Calculus.367_0.6FECEGgqdb67QLM | theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i | Mathlib_Analysis_InnerProductSpace_Calculus |
𝕜 : Type u_1
ι : Type u_2
H : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : Fintype ι
f : H → EuclideanSpace 𝕜 ι
f' : H →L[𝕜] EuclideanSpace 𝕜 ι
t : Set H
y : H
n : ℕ∞
⊢ (∀ (i : ι), ContDiff 𝕜 n fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) ↔ ∀ (i : ι), ContDiff 𝕜 ... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | rfl | theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiff_iff, contDiff_pi]
| Mathlib.Analysis.InnerProductSpace.Calculus.367_0.6FECEGgqdb67QLM | theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ ContDiff ℝ n ↑univUnitBall | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | suffices ContDiff ℝ n fun x : E => (1 + ‖x‖ ^ 2 : ℝ).sqrt⁻¹ from this.smul contDiff_id | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) := by
| Mathlib.Analysis.InnerProductSpace.Calculus.380_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ ContDiff ℝ n fun x => (sqrt (1 + ‖x‖ ^ 2))⁻¹ | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | have h : ∀ x : E, (0 : ℝ) < (1 : ℝ) + ‖x‖ ^ 2 := fun x => by positivity | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) := by
suffices ContDiff ℝ n fun x : E => (1 + ‖x‖ ^ 2 : ℝ).sqrt⁻¹ from this.smul contDiff_id
| Mathlib.Analysis.InnerProductSpace.Calculus.380_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) | Mathlib_Analysis_InnerProductSpace_Calculus |
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
⊢ 0 < 1 + ‖x‖ ^ 2 | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | positivity | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) := by
suffices ContDiff ℝ n fun x : E => (1 + ‖x‖ ^ 2 : ℝ).sqrt⁻¹ from this.smul contDiff_id
have h : ∀ x : E, (0 : ℝ) < (1 : ℝ) + ‖x‖ ^ 2 := fun x => by | Mathlib.Analysis.InnerProductSpace.Calculus.380_0.6FECEGgqdb67QLM | theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) | Mathlib_Analysis_InnerProductSpace_Calculus |
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