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 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : multiplicity p a ≤ multiplicity p b
hab : Coprime a b
⊢ multiplicity p a = 0 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [multiplicity_le_multiplicity_iff] at hle | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
| Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
⊢ multiplicity p a = 0 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity] | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
| Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
⊢ ¬p ^ 1 ∣ a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | intro h | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multipl... | Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
h : p ^ 1 ∣ a
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have := Nat.dvd_gcd h (hle _ h) | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multipl... | Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
h : p ^ 1 ∣ a
this : p ^ 1 ∣ gcd a b
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [Coprime.gcd_eq_one hab, Nat.dvd_one, pow_one] at this | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multipl... | Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
h : p ^ 1 ∣ a
this : p = 1
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact hp this | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multipl... | Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : ¬1 ≤ a
⊢ ¬1 ≤ a * a | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs... | push_neg at h ⊢ | @[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by | Mathlib.Algebra.Order.Monoid.Defs.176_0.P9s7TI8QJTWokXI | @[to_additive (attr | Mathlib_Algebra_Order_Monoid_Defs |
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : a < 1
⊢ a * a < 1 | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs... | exact mul_lt_one' h h | @[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢; | Mathlib.Algebra.Order.Monoid.Defs.176_0.P9s7TI8QJTWokXI | @[to_additive (attr | Mathlib_Algebra_Order_Monoid_Defs |
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : ¬1 < a
⊢ ¬1 < a * a | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs... | push_neg at h ⊢ | @[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by | Mathlib.Algebra.Order.Monoid.Defs.180_0.P9s7TI8QJTWokXI | @[to_additive (attr | Mathlib_Algebra_Order_Monoid_Defs |
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : a ≤ 1
⊢ a * a ≤ 1 | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs... | exact mul_le_one' h h | @[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by push_neg at h ⊢; | Mathlib.Algebra.Order.Monoid.Defs.180_0.P9s7TI8QJTWokXI | @[to_additive (attr | Mathlib_Algebra_Order_Monoid_Defs |
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
⊢ a * a ≤ 1 ↔ a ≤ 1 | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs... | simp [← not_iff_not] | @[to_additive (attr := simp)]
theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 := by | Mathlib.Algebra.Order.Monoid.Defs.184_0.P9s7TI8QJTWokXI | @[to_additive (attr | Mathlib_Algebra_Order_Monoid_Defs |
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
⊢ a * a < 1 ↔ a < 1 | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs... | simp [← not_iff_not] | @[to_additive (attr := simp)]
theorem mul_self_lt_one_iff : a * a < 1 ↔ a < 1 := by | Mathlib.Algebra.Order.Monoid.Defs.187_0.P9s7TI8QJTWokXI | @[to_additive (attr | Mathlib_Algebra_Order_Monoid_Defs |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : Setoid.r a₁ b₁
h₂ : Setoid.r a₂ b₂
⊢ Setoid.r (a₁ * a₂) (b₁ * b₂) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢ | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
| Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
⊢ a₁ * a₂ - b₁ * b₂ ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂) | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
| Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
F : a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ ∈ I
⊢ a₁ * a₂ - b₁ * b₂ ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁] | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right... | Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
F : a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ ∈ I
⊢ a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁] | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right... | Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
F : a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ ∈ I
this : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁
⊢ a₁ * a₂ - b₁ * b₂ ∈... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rwa [← this] at F | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right... | Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX | /-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x✝ y✝ x y : R
⊢ (mk (span {x})) y = 0 ↔ 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, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton] | theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
| Mathlib.RingTheory.Ideal.Quotient.133_0.TwNAv7Pc4PYOWjX | theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x✝ y✝ x y : R
⊢ (mk I) x = (mk I) y ↔ x - y ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [← eq_zero_iff_mem, map_sub, sub_eq_zero] | theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
| Mathlib.RingTheory.Ideal.Quotient.137_0.TwNAv7Pc4PYOWjX | theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
⊢ Subsingleton (R ⧸ I) ↔ I = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem] | theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
| Mathlib.RingTheory.Ideal.Quotient.152_0.TwNAv7Pc4PYOWjX | theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x y : R
⊢ ∀ (a : R ⧸ ⊤), a = default | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨x⟩ | instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by | Mathlib.RingTheory.Ideal.Quotient.157_0.TwNAv7Pc4PYOWjX | instance : Unique (R ⧸ (⊤ : Ideal R)) | Mathlib_RingTheory_Ideal_Quotient |
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x✝ y : R
a✝ : R ⧸ ⊤
x : R
⊢ Quot.mk Setoid.r x = default | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top | instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; | Mathlib.RingTheory.Ideal.Quotient.157_0.TwNAv7Pc4PYOWjX | instance : Unique (R ⧸ (⊤ : Ideal R)) | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
s : Set R
⊢ ⇑(mk I) ⁻¹' (⇑(mk I) '' s) = ⋃ x, (fun y => ↑x + y) '' s | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | ext x | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
| Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s | Mathlib_RingTheory_Ideal_Quotient |
case h
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
I : Ideal R
s : Set R
x : R
⊢ x ∈ ⇑(mk I) ⁻¹' (⇑(mk I) '' s) ↔ x ∈ ⋃ x, (fun y => ↑x + y) '' s | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq] | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
| Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s | Mathlib_RingTheory_Ideal_Quotient |
case h
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
I : Ideal R
s : Set R
x : R
⊢ (∃ x_1 ∈ s, x_1 - x ∈ I) ↔ ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩ | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
... | Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x✝¹ y : R
I : Ideal R
s : Set R
x : R
x✝ : ∃ x_1 ∈ s, x_1 - x ∈ I
a : R
a_in : a ∈ s
h : a - x ∈ I
⊢ ↑{ val := -(a - x), property := (_ : -(a - x) ∈ I) } + a = x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | simp | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
... | Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x✝¹ y : R
I : Ideal R
s : Set R
x : R
x✝ : ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x
i : R
hi : i ∈ I
a : R
ha : a ∈ s
Eq : ↑{ val := i, property := hi } + a = x
⊢ a - x ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub] | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
... | Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x✝¹ y : R
I : Ideal R
s : Set R
x : R
x✝ : ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x
i : R
hi : i ∈ I
a : R
ha : a ∈ s
Eq : ↑{ val := i, property := hi } + a = x
⊢ -↑{ val := i, property := hi } ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact I.neg_mem hi | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
... | Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX | /-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
⊢ IsDomain (R ⧸ I) ↔ IsPrime I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩ | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
| Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime | Mathlib_RingTheory_Ideal_Quotient |
case refine'_1
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
H : IsDomain (R ⧸ I)
⊢ 0 ≠ 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩ | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· | Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime | Mathlib_RingTheory_Ideal_Quotient |
case refine'_1
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
H : IsDomain (R ⧸ I)
this : Nontrivial (R ⧸ I)
⊢ 0 ≠ 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact zero_ne_one | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
| Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime | Mathlib_RingTheory_Ideal_Quotient |
case refine'_2
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
H : IsDomain (R ⧸ I)
x y : R
h : x * y ∈ I
⊢ x ∈ I ∨ y ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢ | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· | Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime | Mathlib_RingTheory_Ideal_Quotient |
case refine'_2
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
H : IsDomain (R ⧸ I)
x y : R
h : (mk I) x * (mk I) y = 0
⊢ (mk I) x = 0 ∨ (mk I) y = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
| Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime | Mathlib_RingTheory_Ideal_Quotient |
case refine'_2
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
H : IsDomain (R ⧸ I)
x y : R
h : (mk I) x * (mk I) y = 0
this : NoZeroDivisors (R ⧸ I)
⊢ (mk I) x = 0 ∨ (mk I) y = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact eq_zero_or_eq_zero_of_mul_eq_zero h | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivi... | Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX | theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
⊢ ∀ {a : R ⧸ I}, a ≠ 0 → ∃ b, a * b = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ h | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
| Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
case mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
⊢ ∃ b, Quot.mk Setoid.r a * b = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩ | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
| Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : b * a + c = 1
⊢ ∃ b, Quot.mk Setoid.r a * b = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [mul_comm] at abc | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
| Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : a * b + c = 1
⊢ ∃ b, Quot.mk Setoid.r a * b = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | refine' ⟨mk _ b, Quot.sound _⟩ | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
| Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : a * b + c = 1
⊢ Setoid.r ((fun x x_1 => x * x_1) a b) 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | simp only [Submodule.quotientRel_r_def] | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
| Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : a * b + c = 1
⊢ a * b - 1 ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [← eq_sub_iff_add_eq'] at abc | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
| Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : c = 1 - a * b
⊢ a * b - 1 ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] ... | Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a : R ⧸ I
ha : a ≠ 0
⊢ (a * if ha : a = 0 then 0 else choose (_ : ∃ b, a * b = 1)) = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [dif_neg ha] | /-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=... | Mathlib.RingTheory.Ideal.Quotient.211_0.TwNAv7Pc4PYOWjX | /-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a : R ⧸ I
ha : a ≠ 0
⊢ a * choose (_ : ∃ b, a * b = 1) = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact Classical.choose_spec (exists_inv ha) | /-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=... | Mathlib.RingTheory.Ideal.Quotient.211_0.TwNAv7Pc4PYOWjX | /-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ IsMaximal I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | apply Ideal.isMaximal_iff.2 | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
| Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ 1 ∉ I ∧ ∀ (J : Ideal R) (x : R), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | constructor | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
| Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case left
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ 1 ∉ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | intro h | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· | Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case left
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
h : 1 ∈ I
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩ | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
| Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case left.intro.mk.intro.mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
hqf : IsField (R ⧸ I)
h : 1 ∈ I
w✝¹ : R ⧸ I
x : R
w✝ : R ⧸ I
y : R
hxy : Quot.mk Setoid.r x ≠ Quot.mk Setoid.r y
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h)) | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
| Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case right
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ ∀ (J : Ideal R) (x : R), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | intro J x hIJ hxnI hxJ | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ ... | Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case right
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
I : Ideal R
hqf : IsField (R ⧸ I)
J : Ideal R
x : R
hIJ : I ≤ J
hxnI : x ∉ I
hxJ : x ∈ J
⊢ 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩ | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ ... | Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case right.intro.mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
hqf : IsField (R ⧸ I)
J : Ideal R
x : R
hIJ : I ≤ J
hxnI : x ∉ I
hxJ : x ∈ J
w✝ : R ⧸ I
y : R
hy : (mk I) x * Quot.mk Setoid.r y = 1
⊢ 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [← zero_add (1 : R), ← sub_self (x * y), sub_add] | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ ... | Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
case right.intro.mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
hqf : IsField (R ⧸ I)
J : Ideal R
x : R
hIJ : I ≤ J
hxnI : x ∉ I
hxJ : x ∈ J
w✝ : R ⧸ I
y : R
hy : (mk I) x * Quot.mk Setoid.r y = 1
⊢ x * y - (x * y - 1) ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy)) | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ ... | Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX | /-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
⊢ Function.Surjective ⇑(lift I f H) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | intro y | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
| Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y✝ : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
y : S
⊢ ∃ a, (lift I f H) a = y | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | obtain ⟨x, rfl⟩ := hf y | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
| Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) | Mathlib_RingTheory_Ideal_Quotient |
case intro
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
x : R
⊢ ∃ a, (lift I f H) a = f x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | use Ideal.Quotient.mk I x | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
| Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) | Mathlib_RingTheory_Ideal_Quotient |
case h
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
x : R
⊢ (lift I f H) ((mk I) x) = f x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | simp only [Ideal.Quotient.lift_mk] | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
| Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX | theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S✝ : Type v
x y : R
inst✝ : CommRing S✝
S T : Ideal R
H : S ≤ T
⊢ RingHom.comp (factor S T H) (mk S) = mk T | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | ext x | @[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
| Mathlib.RingTheory.Ideal.Quotient.290_0.TwNAv7Pc4PYOWjX | @[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T | Mathlib_RingTheory_Ideal_Quotient |
case a
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S✝ : Type v
x✝ y : R
inst✝ : CommRing S✝
S T : Ideal R
H : S ≤ T
x : R
⊢ (RingHom.comp (factor S T H) (mk S)) x = (mk T) x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [RingHom.comp_apply, factor_mk] | @[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
| Mathlib.RingTheory.Ideal.Quotient.290_0.TwNAv7Pc4PYOWjX | @[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T | Mathlib_RingTheory_Ideal_Quotient |
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
src✝ : (R ⧸ I) ≃ₗ[R] R ⧸ J := Submodule.quotEquivOfEq I J h
⊢ ∀ (x y : R ⧸ I),
Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFu... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨x⟩ ⟨y⟩ | /-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
| Mathlib.RingTheory.Ideal.Quotient.298_0.TwNAv7Pc4PYOWjX | /-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
src✝ : (R ⧸ I) ≃ₗ[R] R ⧸ J := Submodule.quotEquivOfEq I J h
x✝ : R ⧸ I
x : R
y✝ : R ⧸ I
y : R
⊢ Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rfl | /-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
| Mathlib.RingTheory.Ideal.Quotient.298_0.TwNAv7Pc4PYOWjX | /-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J | Mathlib_RingTheory_Ideal_Quotient |
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
⊢ RingEquiv.symm (quotEquivOfEq h) = quotEquivOfEq (_ : J = I) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | ext | @[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by | Mathlib.RingTheory.Ideal.Quotient.315_0.TwNAv7Pc4PYOWjX | @[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm | Mathlib_RingTheory_Ideal_Quotient |
case h
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
x✝ : R ⧸ J
⊢ (RingEquiv.symm (quotEquivOfEq h)) x✝ = (quotEquivOfEq (_ : J = I)) x✝ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rfl | @[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; | Mathlib.RingTheory.Ideal.Quotient.315_0.TwNAv7Pc4PYOWjX | @[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
⊢ ∀ (a₁ : R) (a₂ : ι → R) (b₁ : R) (b₂ : ι → R),
Setoid.r a₁ b₁ →
Setoid.r a₂ b₂ →
(fun r m => Submodule.Quotient.mk (r • m)) a₁ a₂ = (fun r m => Submodule.Quotient.mk (r • m)) b₁ b₂ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | intro c₁ m₁ c₂ m₂ hc hm | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
| Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : Setoid.r c₁ c₂
hm : Setoid.r m₁ m₂
⊢ (fun r m => Submodule.Quotient.mk (r • m)) c₁ m₁ = (fun r m => Submodule.Quotient.mk (r • m)) c₂ m₂ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | apply Ideal.Quotient.eq.2 | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
| Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : Setoid.r c₁ c₂
hm : Setoid.r m₁ m₂
⊢ c₁ • m₁ - c₂ • m₂ ∈ pi I ι | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [Submodule.quotientRel_r_def] at hc hm | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
| Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : c₁ - c₂ ∈ I
hm : m₁ - m₂ ∈ pi I ι
⊢ c₁ • m₁ - c₂ • m₂ ∈ pi I ι | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | intro i | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
| Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : c₁ - c₂ ∈ I
hm : m₁ - m₂ ∈ pi I ι
i : ι
⊢ (c₁ • m₁ - c₂ • m₂) i ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact I.mul_sub_mul_mem hc (hm i) | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
| Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (b : (ι → R) ⧸ pi I ι), 1 • b = b | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
b✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ 1 • Quot.mk Setoid.r a = Quot.mk Setoid.r a | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
b✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ (Quotient.mk (pi I ι)) (1 • a) = (Quotient.mk (pi I ι)) a | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | congr with i | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
b✝ : (ι → R) ⧸ pi I ι
a : ι → R
i : ι
⊢ (1 • a) i = a i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact one_mul (a i) | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x y : R ⧸ I) (b : (ι → R) ⧸ pi I ι), (x * y) • b = x • y • b | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quot.mk Setoid.r a * Quot.mk Setoid.r b) • Quot.mk Setoid.r c =
Quot.mk Setoid.r a • Quot.mk Setoid.r b • Quot.mk Setoid.r c | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quotient.mk (pi I ι)) ((fun x x_1 => x * x_1) a b • c) = (Quotient.mk (pi I ι)) (a • b • c) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | congr 1 | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3.h.e_6.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (fun x x_1 => x * x_1) a b • c = a • b • c | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | funext i | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ ((fun x x_1 => x * x_1) a b • c) i = (a • b • c) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact mul_assoc a b (c i) | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (a : R ⧸ I), a • 0 = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b : R
S ι : Type v
a✝ : R ⧸ I
a : R
⊢ Quot.mk Setoid.r a • 0 = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b : R
S ι : Type v
a✝ : R ⧸ I
a : R
⊢ (Quotient.mk (pi I ι)) (a • 0) = (Quotient.mk (pi I ι)) 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | congr with _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : ι
⊢ (a • 0) x✝ = OfNat.ofNat 0 x✝ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact mul_zero a | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (a : R ⧸ I) (x y : (ι → R) ⧸ pi I ι), a • (x + y) = a • x + a • y | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b✝ : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : (ι → R) ⧸ pi I ι
b : ι → R
y✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ Quot.mk Setoid.r a • (Quot.mk Setoid.r b + Quot.mk Setoid.r c) =
Quot.mk Setoid.r a • Quot.mk Setoid.r b + Quot.mk Setoid.r a • Quot.mk Setoid.r c | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b✝ : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : (ι → R) ⧸ pi I ι
b : ι → R
y✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quotient.mk (pi I ι)) (a • (fun x x_1 => x + x_1) b c) =
(Quotient.mk (pi I ι)) ((fun x x_1 => x + x_1) (a • b) (a • c)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | congr with i | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b✝ : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : (ι → R) ⧸ pi I ι
b : ι → R
y✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ (a • (fun x x_1 => x + x_1) b c) i = (fun x x_1 => x + x_1) (a • b) (a • c) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact mul_add a (b i) (c i) | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (r s : R ⧸ I) (x : (ι → R) ⧸ pi I ι), (r + s) • x = r • x + s • x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quot.mk Setoid.r a + Quot.mk Setoid.r b) • Quot.mk Setoid.r c =
Quot.mk Setoid.r a • Quot.mk Setoid.r c + Quot.mk Setoid.r b • Quot.mk Setoid.r c | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quotient.mk (pi I ι)) ((fun x x_1 => x + x_1) a b • c) =
(Quotient.mk (pi I ι)) ((fun x x_1 => x + x_1) (a • c) (b • c)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | congr with i | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ ((fun x x_1 => x + x_1) a b • c) i = (fun x x_1 => x + x_1) (a • c) (b • c) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact add_mul a b (c i) | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x : (ι → R) ⧸ pi I ι), 0 • x = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨a⟩ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ 0 • Quot.mk Setoid.r a = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _ | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ (Quotient.mk (pi I ι)) (0 • a) = (Quotient.mk (pi I ι)) 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | congr with i | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
i : ι
⊢ (0 • a) i = OfNat.ofNat 0 i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact zero_mul (a i) | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact... | Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨_⟩ ⟨_⟩ | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a✝¹ : ι → R
y✝ : (ι → R) ⧸ pi I ι
a✝ : ι → R
⊢ (fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rfl | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (r : R ⧸ I) (x : (ι → R) ⧸ pi I ι),
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i =>... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨_⟩ ⟨_⟩ | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
case mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
r✝ : R ⧸ I
a✝¹ : R
x✝ : (ι → R) ⧸ pi I ι
a✝ : ι → R
⊢ AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (f... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rfl | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ Function.LeftInverse (fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i))
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rintro ⟨x⟩ | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
x : ι → R
⊢ (fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i))
(AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i =>... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact Ideal.Quotient.eq.2 fun i => Ideal.Quotient.eq.1 (Quotient.out_eq' _) | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ Function.RightInverse (fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i))
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | intro x | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x : ι → R ⧸ I
⊢ AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | ext i | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
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