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 |
|---|---|---|---|---|---|---|
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val := p₂, proper... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simpa using hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂)) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup' | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y
⊢ ((p : { x //... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | refine' MulEquiv.trans (N := ∀ p : ps, P p) _ _ | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_1
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | show (∀ p : ps, ∀ P : Sylow p G, P) ≃* ∀ p : ps, P p | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_1
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply @MulEquiv.piCongrRight ps (fun p => ∀ P : Sylow p G, P) (fun p => P p) _ _ | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_1
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rintro ⟨p, hp⟩ | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_1.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_1.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | letI := unique_of_normal _ (hn (P p)) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_1.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply MulEquiv.piUnique | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | show (∀ p : ps, P p) ≃* G | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply (bijective_iff_injective_and_card _).mpr | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | constructor | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commut... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | show Injective _ | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commut... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply Subgroup.injective_noncommPiCoprod_of_independent | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → C... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply independent_of_coprime_order hcomm | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → C... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | have hne' : p₁ ≠ p₂ := by simpa using hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y
p₁ : ℕ
hp₁ : p... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | simpa using hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | apply IsPGroup.coprime_card_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup' | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.right
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commu... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | show card (∀ p : ps, P p) = card G | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case refine'_2.right
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commu... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | calc
card (∀ p : ps, P p) = ∏ p : ps, card (P p) := Fintype.card_pi
_ = ∏ p : ps, p.1 ^ (card G).factorization p.1 := by
congr 1 with ⟨p, hp⟩
exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_primeFactors hp⟩ (P p)
_ = ∏ p in ps, p ^ (card G).factorization p :=
(Finset.prod... | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y
⊢ ∏ p : { x //... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | congr 1 with ⟨p, hp⟩ | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
case e_f.h.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
hcomm : _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_primeFactors hp⟩ (P p) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Sub... | Mathlib_GroupTheory_Sylow |
R : CommRingCat
⊢ IsIso (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away 1))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.... | cases R | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
| Mathlib.Algebra.Category.Ring.Instances.24_0.KsJUUT2FWBN0k2J | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) | Mathlib_Algebra_Category_Ring_Instances |
case mk
α✝ : Type ?u.2113
str✝ : CommRing α✝
⊢ IsIso (CommRingCat.ofHom (algebraMap (↑(Bundled.mk α✝)) (Localization.Away 1))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.... | exact localization_unit_isIso _ | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
| Mathlib.Algebra.Category.Ring.Instances.24_0.KsJUUT2FWBN0k2J | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) | Mathlib_Algebra_Category_Ring_Instances |
R : CommRingCat
M : Submonoid ↑R
⊢ Epi (CommRingCat.ofHom (algebraMap (↑R) (Localization M))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.... | rcases R with ⟨α, str⟩ | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
| Mathlib.Algebra.Category.Ring.Instances.40_0.KsJUUT2FWBN0k2J | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) | Mathlib_Algebra_Category_Ring_Instances |
case mk
α : Type ?u.6221
str : CommRing α
M : Submonoid ↑(Bundled.mk α)
⊢ Epi (CommRingCat.ofHom (algebraMap (↑(Bundled.mk α)) (Localization M))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.... | exact IsLocalization.epi M _ | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
rcases R with ⟨α, str⟩
| Mathlib.Algebra.Category.Ring.Instances.40_0.KsJUUT2FWBN0k2J | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) | Mathlib_Algebra_Category_Ring_Instances |
R S : CommRingCat
f : R ≅ S
a : ↑R
ha : IsUnit (f.hom a)
⊢ IsUnit a | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.... | convert f.inv.isUnit_map ha | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R ≅ S) : IsLocalRingHom f.hom :=
{ map_nonunit := fun a ha => by
| Mathlib.Algebra.Category.Ring.Instances.52_0.KsJUUT2FWBN0k2J | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R ≅ S) : IsLocalRingHom f.hom | Mathlib_Algebra_Category_Ring_Instances |
case h.e'_3
R S : CommRingCat
f : R ≅ S
a : ↑R
ha : IsUnit (f.hom a)
⊢ a = f.inv (f.hom a) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.... | exact (RingHom.congr_fun f.hom_inv_id _).symm | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R ≅ S) : IsLocalRingHom f.hom :=
{ map_nonunit := fun a ha => by
convert f.inv.isUnit_map ha
| Mathlib.Algebra.Category.Ring.Instances.52_0.KsJUUT2FWBN0k2J | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R ≅ S) : IsLocalRingHom f.hom | Mathlib_Algebra_Category_Ring_Instances |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ (coevaluation K V) 1 =
let bV := Basis.ofVectorSpace K V;
∑ i : ↑(Basis.ofVectorSpaceIndex K V), bV i ⊗ₜ[K] Basis.coord bV i | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [coevaluation, id] | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by
| Mathlib.LinearAlgebra.Coevaluation.48_0.2OSHLJKAlhD35xC | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ ((Basis.constr (Basis.singleton Unit K) K) fun x =>
∑ i : ↑(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) i ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) i)
1 =
∑ i : ↑... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [(Basis.singleton Unit K).constr_apply_fintype K] | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by
simp only [coevaluation, id]
| Mathlib.LinearAlgebra.Coevaluation.48_0.2OSHLJKAlhD35xC | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ ∑ i : Unit,
(Basis.equivFun (Basis.singleton Unit K)) 1 i •
∑ i : ↑(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) i ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) i =
∑ ... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton] | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
| Mathlib.LinearAlgebra.Coevaluation.48_0.2OSHLJKAlhD35xC | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ
↑(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ∘ₗ
LinearMap.lTensor (Module.Dual K V) (coevaluatio... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | letI := Classical.decEq (Basis.ofVectorSpaceIndex K V) | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ
↑(LinearEquiv.symm (TensorProduct.assoc... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | apply TensorProduct.ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ LinearMap.compr₂ (mk K (Module.Dual K V) K)
(LinearMap.rTensor (Module.Dual K V) (contract... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | apply (Basis.ofVectorSpace K V).dualBasis.ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∀ (i : ↑(Basis.ofVectorSpaceIndex K V)),
(LinearMap.compr₂ (mk K (Module.Dual K V) K)
... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | intro j | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.compr₂ (mk K (Module.Dual K V) K)
(Linear... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | apply LinearMap.ext_ring | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ((LinearMap.compr₂ (mk K (Module.Dual K V) K)
(L... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [rid_tmul, one_smul, lid_symm_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [TensorProduct.tmul_sum, map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [assoc_symm_tmul] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),
(LinearMap.rT... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [LinearMap.rTensor_tmul, contractLeft_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),
((Basis.dualB... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (∑ x : ↑(Basis.ofVectorSpaceIndex K V), if x = j then 1 ⊗ₜ... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [Finset.sum_ite_eq'] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (if j ∈ Finset.univ then 1 ⊗ₜ[K] Basis.coord (Basis.ofVect... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [Finset.mem_univ, if_true] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) ... | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ LinearMap.lTensor V (contractLeft K V) ∘ₗ
↑(TensorProduct.assoc K V (Module.Dual K V) V) ∘ₗ LinearMap.rTensor V (coevaluation K V) =
↑(LinearEquiv.symm (TensorProduct.rid K V)) ∘ₗ ↑(TensorProduct... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | letI := Classical.decEq (Basis.ofVectorSpaceIndex K V) | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ LinearMap.lTensor V (contractLeft K V) ∘ₗ
↑(TensorProduct.assoc K V (Module.Dual K V) V) ∘ₗ Linea... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | apply TensorProduct.ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ LinearMap.compr₂ (mk K K V)
(LinearMap.lTensor V (contractLeft K V) ∘ₗ
↑(TensorPro... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | apply LinearMap.ext_ring | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.compr₂ (mk K K V)
(LinearMap.lTensor V (contractLeft K V) ∘ₗ
↑(Te... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | apply (Basis.ofVectorSpace K V).ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∀ (i : ↑(Basis.ofVectorSpaceIndex K V)),
((LinearMap.compr₂ (mk K K V)
(Linear... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | intro j | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ((LinearMap.compr₂ (mk K K V)
(LinearMap.lTensor... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V) ∘ₗ
↑(Tenso... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
((TensorPro... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [lid_tmul, one_smul, rid_symm_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
((TensorPro... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
((TensorPro... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [TensorProduct.sum_tmul, map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
(∑ x : ↑(Ba... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [assoc_tmul] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
(∑ x : ↑(Ba... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),
(LinearMap.lT... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [LinearMap.lTensor_tmul, contractLeft_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),
(Basis.ofVect... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (∑ x : ↑(Basis.ofVectorSpaceIndex K V), if j = x then (Bas... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | rw [Finset.sum_ite_eq] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (if j ∈ Finset.univ then (Basis.ofVectorSpace K V) j ⊗ₜ[K]... | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | simp only [Finset.mem_univ, if_true] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _... | Mathlib_LinearAlgebra_Coevaluation |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
⊢ HasStrictFDerivAt f ((continuousM... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _) | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
⊢ (fun y => ‖y - (x, x)‖ * ‖y.1 - y... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | refine' isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, _, EventuallyEq.rfl⟩ | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
| Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
⊢ Tendsto (fun y => ‖y - (x, x)‖) (... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _ | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, _, Eventual... | Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
⊢ ‖id (x, x) - (x, x)‖ = 0 | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rw [_root_.id, sub_self, norm_zero] | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, _, Eventual... | Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
⊢ Ha... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | apply A.congr | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | intro z hz | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | dsimp | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rw [← h.fderiv_eq, add_sub_cancel'_right] | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case A
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case A
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact
(continuousMultilinearCurryFin1 𝕜 E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case B
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f)
((continuousMultilinearCurryFin1 𝕜 E F :
(E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F).compFormalMu... | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
⊢ AnalyticOn 𝕜 ... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | intro y hy | /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (fderiv 𝕜 f) s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.118_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (fderiv 𝕜 f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
y : E
hy : y ∈ s... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rcases h y hy with ⟨p, r, hp⟩ | /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (fderiv 𝕜 f) s := by
intro y hy
| Mathlib.Analysis.Calculus.FDeriv.Analytic.118_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (fderiv 𝕜 f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p✝ : FormalMultilinearSeries 𝕜 E F
r✝ : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact hp.fderiv.analyticAt | /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (fderiv 𝕜 f) s := by
intro y hy
rcases h y hy with ⟨p, r, hp⟩
| Mathlib.Analysis.Calculus.FDeriv.Analytic.118_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (fderiv 𝕜 f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ
⊢ Analytic... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | induction' n with n IH | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
⊢ Anal... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rw [iteratedFDeriv_zero_eq_comp] | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
⊢ Anal... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact ((continuousMultilinearCurryFin0 𝕜 E F).symm : F →L[𝕜] E[×0]→L[𝕜] F).comp_analyticOn h | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rw [iteratedFDeriv_succ_eq_comp_left] | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinear... | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | convert @ContinuousLinearMap.comp_analyticOn 𝕜 E
?_ (ContinuousMultilinearMap 𝕜 (fun _ : Fin (n + 1) ↦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinear... | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case h.e'_9.h.e'_4
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | case g =>
exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F) | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinear... | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ
IH : Analy... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | case g =>
exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F) | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinear... | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ
IH : Analy... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F) | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinear... | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case h.e'_9.h.e'_4
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rfl | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
induction' n with n IH
· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinear... | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ∞
⊢ ContDif... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | let t := { x | AnalyticAt 𝕜 f x } | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ∞
t : Set E... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | suffices : ContDiffOn 𝕜 n f t | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ∞
t : Set E... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact this.mono h | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ∞... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | have H : AnalyticOn 𝕜 f t := fun x hx => hx | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ∞... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | have t_open : IsOpen t := isOpen_analyticAt 𝕜 f | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ∞... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | apply contDiffOn_of_continuousOn_differentiableOn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hcont
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rintro m - | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
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