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 |
|---|---|---|---|---|---|---|
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f g : CentroidHom α
src✝ : α →+ α := ↑f - ↑g
a b : α
⊢ (⇑f - ⇑g) (a * b) = (⇑f - ⇑g) a * b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [map_mul_right, sub_mul] | instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b
| Mathlib.Algebra.Ring.CentroidHom.514_0.FQQ3LT1tg3cKlkH | instance : Sub (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalRing α
h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0
src✝ : Ring (CentroidHom α) := instRing
f g : CentroidHom α
⊢ f * g = g * f | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | ext | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
| Mathlib.Algebra.Ring.CentroidHom.591_0.FQQ3LT1tg3cKlkH | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
case h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalRing α
h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0
src✝ : Ring (CentroidHom α) := instRing
f g : CentroidHom α
a✝ : α
⊢ (f * g) a✝ = (g * f) a✝ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | refine' sub_eq_zero.1 (or_self_iff.1 <| (h _ _) fun r ↦ _) | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
ext
| Mathlib.Algebra.Ring.CentroidHom.591_0.FQQ3LT1tg3cKlkH | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
case h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalRing α
h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0
src✝ : Ring (CentroidHom α) := instRing
f g : CentroidHom α
a✝ r : α
⊢ ((f * g) a✝ - (g * f) a✝) * r * ((f * g) a✝ - (g * f) a✝) = 0 | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [mul_assoc, sub_mul, sub_eq_zero, ← map_mul_right, ← map_mul_right, coe_mul, coe_mul,
comp_mul_comm] | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
ext
refine' sub_eq_zero.1 (or_self_iff.1 <| (h _ _) fun r ↦ _)
| Mathlib.Algebra.Ring.CentroidHom.591_0.FQQ3LT1tg3cKlkH | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
β : Type u_1
G : Type u_2
α : Type u_3
γ : Type u_4
inst✝³ : Group G
inst✝² : AddGroup α
inst✝¹ : SMul γ α
inst✝ : SlashAction β G α γ
k : β
g : G
a : α
⊢ (-a) ∣[k;γ] g + a ∣[k;γ] g = 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash] | @[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
| Mathlib.NumberTheory.ModularForms.SlashActions.58_0.3orIHeXinm1hkfX | @[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g | Mathlib_NumberTheory_ModularForms_SlashActions |
R : Type u_1
β : Type u_2
G : Type u_3
α : Type u_4
γ : Type u_5
inst✝⁷ : Group G
inst✝⁶ : AddGroup α
inst✝⁵ : Monoid γ
inst✝⁴ : MulAction γ α
inst✝³ : SMul R γ
inst✝² : SMul R α
inst✝¹ : IsScalarTower R γ α
inst✝ : SlashAction β G α γ
k : β
g : G
a : α
r : R
⊢ (r • a) ∣[k;γ] g = r • a ∣[k;γ] g | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul] | @[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
| Mathlib.NumberTheory.ModularForms.SlashActions.65_0.3orIHeXinm1hkfX | @[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g | Mathlib_NumberTheory_ModularForms_SlashActions |
β : Type u_1
G : Type u_2
H : Type u_3
α : Type u_4
γ : Type u_5
inst✝⁴ : Group G
inst✝³ : AddMonoid α
inst✝² : SMul γ α
inst✝¹ : Group H
inst✝ : SlashAction β G α γ
h : H →* G
k : β
a : α
⊢ (fun k g => SlashAction.map γ k (h g)) k 1 a = a | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [map_one, SlashAction.slash_one] | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := b... | Mathlib.NumberTheory.ModularForms.SlashActions.75_0.3orIHeXinm1hkfX | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g | Mathlib_NumberTheory_ModularForms_SlashActions |
β : Type u_1
G : Type u_2
H : Type u_3
α : Type u_4
γ : Type u_5
inst✝⁴ : Group G
inst✝³ : AddMonoid α
inst✝² : SMul γ α
inst✝¹ : Group H
inst✝ : SlashAction β G α γ
h : H →* G
k : β
g gg : H
a : α
⊢ (fun k g => SlashAction.map γ k (h g)) k (g * gg) a =
(fun k g => SlashAction.map γ k (h g)) k gg ((fun k g => Slash... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [map_mul, SlashAction.slash_mul] | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := b... | Mathlib.NumberTheory.ModularForms.SlashActions.75_0.3orIHeXinm1hkfX | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
⊢ f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ext1 x | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
⊢ (f ∣[k](A * B)) x = ((f ∣[k]A) ∣[k]B) x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp_rw [slash, UpperHalfPlane.denom_cocycle A B x] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
⊢ f ((A * B) • x) * ↑(Matrix.det ↑↑(A * B)) ^ (k - 1) * (denom A (smulAux B x) * denom B x) ^ (-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A (B • x) ^ (-k) * ↑(Matrix.det ↑↑B) ^ (k - 1) *
denom B x ^ (-k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
⊢ (A * B) • x = A • B • x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | convert UpperHalfPlane.mul_smul' A B x | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by | Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f ((A * B) • x) * ↑(Matrix.det ↑↑(A * B)) ^ (k - 1) * (denom A (smulAux B x) * denom B x) ^ (-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A (B • x) ^ (-k) * ↑(Matrix.det ↑↑B) ^ (k - 1) *... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [e3] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f (A • B • x) * ↑(Matrix.det ↑↑(A * B)) ^ (k - 1) * (denom A (smulAux B x) * denom B x) ^ (-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A (B • x) ^ (-k) * ↑(Matrix.det ↑↑B) ^ (k - 1) *
... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at * | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f (A • B • x) * (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) *
((↑(↑↑A 1 0) * ((↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1))) + ↑(↑↑A 1 1)) *
(↑(↑↑B 1 0) * ↑x + ↑(↑... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | field_simp | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofR... | Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f (A • B • x) * (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) /
((↑(↑↑A 1 0) * (↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) + ↑(↑↑A 1 1)) *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_... | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofR... | Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) = ↑(Matrix.det ↑↑A) ^ (k - 1) * ↑(Matrix.det ↑↑B) ^ (k - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp_rw [← mul_zpow] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofR... | Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
this : (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) = ↑(Matrix.det ↑↑A) ^ (k - 1) * ↑(Matrix.det ↑↑B) ^ (k - 1)
⊢ f (A • B • x) * (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) /
((↑(↑↑A 1 0)... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp_rw [this, ← mul_assoc, ← mul_zpow] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofR... | Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
⊢ (f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ext1 | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.119_0.3orIHeXinm1hkfX | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ ((f + g) ∣[k]A) x✝ = (f ∣[k]A + g ∣[k]A) x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [slash, Pi.add_apply, denom, zpow_neg] | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
| Mathlib.NumberTheory.ModularForms.SlashActions.119_0.3orIHeXinm1hkfX | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ (f (A • x✝) + g (A • x✝)) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ =
f (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ +
g (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k ... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ring | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
| Mathlib.NumberTheory.ModularForms.SlashActions.119_0.3orIHeXinm1hkfX | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
f : ℍ → ℂ
⊢ ∀ (x : ℍ), (f ∣[k]1) x = f x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp [slash, denom] | private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by | Mathlib.NumberTheory.ModularForms.SlashActions.125_0.3orIHeXinm1hkfX | private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
⊢ (c • f) ∣[k]A = c • f ∣[k]A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)] | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
⊢ ((c • 1) • f) ∣[k]A = (c • 1) • f ∣[k]A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ext1 | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
x✝ : ℍ
⊢ (((c • 1) • f) ∣[k]A) x✝ = ((c • 1) • f ∣[k]A) x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp_rw [slash] | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
x✝ : ℍ
⊢ ((c • 1) • f) (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A x✝ ^ (-k) = ((c • 1) • f ∣[k]A) x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply] | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
x✝ : ℍ
⊢ c • 1 * f (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A x✝ ^ (-k) =
c • 1 * (f (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A x✝ ^ (-k)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ring | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
x✝ : ℍ
⊢ (0 ∣[k]A) x✝ = OfNat.ofNat 0 x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [slash, Pi.zero_apply, zero_mul] | private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by | Mathlib.NumberTheory.ModularForms.SlashActions.138_0.3orIHeXinm1hkfX | private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
⊢ 1 ∣[0] A = 1 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe' | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
this : Matrix.det ↑↑↑A = 1
⊢ 1 ∣[0] A = 1 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | funext | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
this : Matrix.det ↑↑↑A = 1
x✝ : ℍ
⊢ (1 ∣[0] A) x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [SL_slash, slash_def, slash, zero_sub, this] | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
this : Matrix.det ↑↑↑A = 1
x✝ : ℍ
⊢ OfNat.ofNat 1 (↑A • x✝) * ↑1 ^ (-1) * denom (↑A) x✝ ^ (-0) = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_sl... | Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ (f ∣[k] γ) z = f z ↔ f (γ • z) = (↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k * f z | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [subgroup_slash, slash_def, ModularForm.slash] | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ f (↑↑γ • z) * ↑(Matrix.det ↑↑↑↑γ) ^ (k - 1) * denom (↑↑γ) z ^ (-k) = f z ↔
f (γ • z) = (↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k * f z | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2 | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib_NumberTheory_ModularForms_SlashActions |
case h.e'_1.h.e'_2
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ f (↑↑γ • z) * ↑(Matrix.det ↑↑↑↑γ) ^ (k - 1) * denom (↑↑γ) z ^ (-k) =
((↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k)⁻¹ * f (γ • z) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [mul_comm] | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib_NumberTheory_ModularForms_SlashActions |
case h.e'_1.h.e'_2
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ denom (↑↑γ) z ^ (-k) * (f (↑↑γ • z) * ↑(Matrix.det ↑↑↑↑γ) ^ (k - 1)) =
((↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k)⁻¹ * f (γ • z) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb] | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib_NumberTheory_ModularForms_SlashActions |
case h.e'_1.h.e'_2
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ ((↑(↑↑↑↑γ 1 0) * ↑z + ↑(↑↑↑↑γ 1 1)) ^ k)⁻¹ * f (↑↑γ • z) = ((↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k)⁻¹ * f (↑↑γ • z) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rfl | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib_NumberTheory_ModularForms_SlashActions |
case convert_4
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ (↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k ≠ 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | convert zpow_ne_zero k (denom_ne_zero γ z) | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
... | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
⊢ (f * g) ∣[k1 + k2] A = Matrix.det ↑↑A • f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ext1 x | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
⊢ ((f * g) ∣[k1 + k2] A) x = (Matrix.det ↑↑A • f ∣[k1] A * g ∣[k2] A) x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
⊢ f (A • x) * g (A • x) * ↑(Matrix.det ↑↑A) ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
↑(Matrix.det ↑↑A) * (f (A • x) * ↑(Matrix.det ↑↑A) ^ (k1 - 1) * denom A x ^ (-k1)) *
(g (A • x) * ↑(Matrix.det ↑↑A) ^ (k2 - 1) * den... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | set d : ℂ := ↑((↑ₘA).det : ℝ) | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ d ≠ 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | dsimp | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ ¬↑(Matrix.det ↑↑A) = 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | norm_cast | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ ¬Matrix.det ↑↑A = 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | exact Matrix.GLPos.det_ne_zero A | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
this : d ≠ 0
⊢ d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [← zpow_one_add₀ this, ← zpow_add₀ this] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
this : d ≠ 0
⊢ d ^ (k1 + k2 - 1) = d ^ (1 + (k1 - 1) + (k2 - 1)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | congr | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case e_a
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
this : d ≠ 0
⊢ k1 + k2 - 1 = 1 + (k1 - 1) + (k2 - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ring | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ ... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
⊢ denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [Int.neg_add, zpow_add₀] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case ha
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
⊢ denom A x ≠ 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | exact UpperHalfPlane.denom_ne_zero A x | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2)
⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [h1, h22] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2)
⊢ f (A • x) * g (A • x) * (d * d ^ (k1 - 1) * d ^ (k2 - 1)) * (denom A x ^ (... | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | ring | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
ha... | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : SL(2, ℤ)
f g : ℍ → ℂ
⊢ (f * g) ∣[k1 + k2] ↑A = Matrix.det ↑↑↑A • f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | apply mul_slash | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.223_0.3orIHeXinm1hkfX | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : SL(2, ℤ)
f g : ℍ → ℂ
⊢ Matrix.det ↑↑↑A • f ∣[k1] A * g ∣[k2] A = 1 • f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [det_coe'] | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
apply mul_slash
_ = (1 : ℝ) • f ∣[k1] A * g ∣[k2] A := by | Mathlib.NumberTheory.ModularForms.SlashActions.223_0.3orIHeXinm1hkfX | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : SL(2, ℤ)
f g : ℍ → ℂ
⊢ 1 • f ∣[k1] A * g ∣[k2] A = f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#a... | rw [one_smul] | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
apply mul_slash
_ = (1 : ℝ) • f ∣[k1] A * g ∣[k2] A := by rw [det_coe']
_ = f ∣[k1] A *... | Mathlib.NumberTheory.ModularForms.SlashActions.223_0.3orIHeXinm1hkfX | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P✝ : Type u_3
inst✝ : CommSemiring P✝
P : Ideal R
hp : IsPrime P
⊢ 1 ∈ { carrier := (↑P)ᶜ, mul_mem' := (_ : ∀ {x y : R}, x ∈ (↑P)ᶜ → y ∈ (↑P)ᶜ → x * y ∈ ↑P → False) }.carrier | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | convert P.ne_top_iff_one.1 hp.1 | /-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by | Mathlib.RingTheory.Localization.AtPrime.45_0.QSwWrbtcZl7L7lq | /-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : 0 = 1
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
t : ↥(Ideal.primeCompl P)
ht : ↑t * 0 = ↑t * ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | have htz : (t : R) = 0 := by simpa using ht.symm | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
t : ↥(Ideal.primeCompl P)
ht : ↑t * 0 = ↑t * 1
⊢ ↑t = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | simpa using ht.symm | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by | Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
t : ↥(Ideal.primeCompl P)
ht : ↑t * 0 = ↑t * ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P) | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this : _root_.Nontrivial S := Nontrivial S P
⊢ ∀ (a b : S), a ∈ nonunits S → b ∈ nonunits S → a + b ∈ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | intro x y hx hy hu | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUn... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | cases' isUnit_iff_exists_inv.1 hu with z hxyz | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonu... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
h... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
h... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [← hrx] at hx | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
h... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [← hry] at hy | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
h... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | suffices : (t : R) * (sx * sy * sz) ∈ P | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝¹ : _root_.Nontrivial S := Nontrivial S P
x ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2) | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case this
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [← ht] | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case this
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have... | Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁶ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
P : Type u_3
inst✝³ : CommSemiring P
A : Type u_4
inst✝² : CommRing A
inst✝¹ : IsDomain A
I : Ideal R
hI : Ideal.IsPrime I
inst✝ : IsLocalization.AtPrime S I
x : R
h : optParam (LocalRing S) (_ : LocalRing S)
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x | theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
| Mathlib.RingTheory.Localization.AtPrime.145_0.QSwWrbtcZl7L7lq | theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁶ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
P : Type u_3
inst✝³ : CommSemiring P
A : Type u_4
inst✝² : CommRing A
inst✝¹ : IsDomain A
I : Ideal R
hI : Ideal.IsPrime I
inst✝ : IsLocalization.AtPrime S I
h : optParam (LocalRing S) (_ : LocalRing S)
x : R
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x | theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by | Mathlib.RingTheory.Localization.AtPrime.152_0.QSwWrbtcZl7L7lq | theorem comap_maximalIdeal (h : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁶ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
P : Type u_3
inst✝³ : CommSemiring P
A : Type u_4
inst✝² : CommRing A
inst✝¹ : IsDomain A
I : Ideal R
hI : Ideal.IsPrime I
inst✝ : IsLocalization.AtPrime S I
x : R
y : ↥(Ideal.primeCompl I)
h : optParam (Local... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y | theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
| Mathlib.RingTheory.Localization.AtPrime.162_0.QSwWrbtcZl7L7lq | theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
⊢ Ideal.map (algebraMap R (Localization.AtPrime I)) I = LocalRing.maximalIdeal (Localization (Ideal.primeCompl I)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localizat... | Mathlib.RingTheory.Localization.AtPrime.192_0.QSwWrbtcZl7L7lq | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localizat... | Mathlib_RingTheory_Localization_AtPrime |
case h.e'_3
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
⊢ LocalRing.maximalIdeal (Localization (Ideal.primeCompl I)) =
Ideal.map (algebraMap R (Localization.AtPrime I))
(Idea... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [map_comap I.primeCompl] | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localizat... | Mathlib.RingTheory.Localization.AtPrime.192_0.QSwWrbtcZl7L7lq | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localizat... | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
h : Ideal.primeCompl I ≤ Submonoid.comap f (Ideal.primeCompl J)
x : R
hx : x ∈ Ideal.comap f J
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | contrapose! hx | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
| Mathlib.RingTheory.Localization.AtPrime.204_0.QSwWrbtcZl7L7lq | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
h : Ideal.primeCompl I ≤ Submonoid.comap f (Ideal.primeCompl J)
x : R
hx : x ∉ I
⊢ x ∉ Ideal.co... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | exact h hx | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
| Mathlib.RingTheory.Localization.AtPrime.204_0.QSwWrbtcZl7L7lq | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
x : Localization.AtPrime I
hx : IsUnit ((localRingHom I J f hIJ) x)
⊢... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩ | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
r : R
s : ↥(Ideal.primeCompl I)
hx : IsUnit ((localR... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [localRingHom_mk'] at hx | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
r : R
s : ↥(Ideal.primeCompl I)
hx :
IsUnit
(m... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [AtPrime.isUnit_mk'_iff] at hx ⊢ | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
r : R
s : ↥(Ideal.primeCompl I)
hx : f r ∈ Ideal.pri... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | exact fun hr => hx ((SetLike.ext_iff.mp hIJ r).mp hr) | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
rw [AtPrime.isUnit_mk'_... | Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S✝ : Type u_2
inst✝³ : CommSemiring S✝
inst✝² : Algebra R S✝
P : Type u_3
inst✝¹ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
S : Type u_4
inst✝ : CommSemiring S
J : Ideal S
hJ : Ideal.IsPrime J
K : Ideal P
hK : Ideal.IsPrime K
f : R →+* S
hIJ : I = Ideal.comap ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | rw [hIJ, hJK, Ideal.comap_comap f g] | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by | Mathlib.RingTheory.Localization.AtPrime.259_0.QSwWrbtcZl7L7lq | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom... | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S✝ : Type u_2
inst✝³ : CommSemiring S✝
inst✝² : Algebra R S✝
P : Type u_3
inst✝¹ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
S : Type u_4
inst✝ : CommSemiring S
J : Ideal S
hJ : Ideal.IsPrime J
K : Ideal P
hK : Ideal.IsPrime K
f : R →+* S
hIJ : I = Ideal.comap ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theor... | simp only [Function.comp_apply, RingHom.coe_comp, localRingHom_to_map] | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom... | Mathlib.RingTheory.Localization.AtPrime.259_0.QSwWrbtcZl7L7lq | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom... | Mathlib_RingTheory_Localization_AtPrime |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j : ι
h : ¬ComplexShape.Rel c i j
⊢ (fun i j => (d C i j).app t) i j = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | have := C.shape _ _ h | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j : ι
h : ¬ComplexShape.Rel c i j
this : d C i j = 0
⊢ (fun i j => (d C i j).app t) i j = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | rw [NatTrans.ext_iff, Function.funext_iff] at this | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j : ι
h : ¬ComplexShape.Rel c i j
this : ∀ (a : T), (d C i j).app a = 0.app a
⊢ (fun i j => (d C i j).app t) i j = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | exact this t | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j k : ι
x✝¹ : ComplexShape.Rel c i j
x✝ : ComplexShape.Rel c j k
⊢ (fun i j => (d C i j).app t) i j ≫ (fun i j => (d C i j).app t) j k =... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | have := C.d_comp_d i j k | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j k : ι
x✝¹ : ComplexShape.Rel c i j
x✝ : ComplexShape.Rel c j k
this : d C i j ≫ d C j k = 0
⊢ (fun i j => (d C i j).app t) i j ≫ (fun ... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | rw [NatTrans.ext_iff, Function.funext_iff] at this | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j k : ι
x✝¹ : ComplexShape.Rel c i j
x✝ : ComplexShape.Rel c j k
this : ∀ (a : T), (d C i j ≫ d C j k).app a = 0.app a
⊢ (fun i j => (d ... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | exact this t | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
⊢ { obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalCo... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | ext i | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i : ι
⊢ Hom.f
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Ho... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | dsimp | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i : ι
⊢ (X C i).map (𝟙 t) = 𝟙 ((X C i).obj t) | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | rw [(C.X i).map_id] | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
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