state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case refine_1
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → Ideal R
x : ι → R
a✝ : ∀ i ∈ ∅, x i ∈ I i
⊢ ∏ i in ∅, x i ∈ ∏ i in ∅, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
refine Finset.induction_on s ?_ ?_
· intro
| Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
case refine_1
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → Ideal R
x : ι → R
a✝ : ∀ i ∈ ∅, x i ∈ I i
⊢ 1 ∈ ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact Submodule.mem_top | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
refine Finset.induction_on s ?_ ?_
· intro
rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]
| Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
case refine_2
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → Ideal R
x : ι → R
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
a ∉ s →
((∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i) →
(∀ i ∈ insert a s, x i ∈ I i) → ∏ i in insert a s, x i ∈ ∏ i in insert a s, ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro a s ha IH h | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
refine Finset.induction_on s ?_ ?_
· intro
rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]
exact Submodule.mem_top
· | Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
case refine_2
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s✝ : Finset ι
I : ι → Ideal R
x : ι → R
a : ι
s : Finset ι
ha : a ∉ s
IH : (∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i
h : ∀ i ∈ insert a s, x i ∈ I i
⊢ ∏ i in insert a s, x i ∈ ∏ i in insert a s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Finset.prod_insert ha, Finset.prod_insert ha] | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
refine Finset.induction_on s ?_ ?_
· intro
rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]
exact Submodule.mem_top
· intro a s ha IH... | Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
case refine_2
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s✝ : Finset ι
I : ι → Ideal R
x : ι → R
a : ι
s : Finset ι
ha : a ∉ s
IH : (∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i
h : ∀ i ∈ insert a s, x i ∈ I i
⊢ x a * ∏ x_1 in s, x x_1 ∈ I a * ∏ x in s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact
mul_mem_mul (h a <| Finset.mem_insert_self a s)
(IH fun i hi => h i <| Finset.mem_insert_of_mem hi) | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
refine Finset.induction_on s ?_ ?_
· intro
rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]
exact Submodule.mem_top
· intro a s ha IH... | Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
S T : Set R
⊢ span S * span T = span (S * T) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | unfold span | theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by
| Mathlib.RingTheory.Ideal.Operations.522_0.5qK551sG47yBciY | theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
S T : Set R
⊢ Submodule.span R S * Submodule.span R T = Submodule.span R (S * T) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Submodule.span_mul_span] | theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by
unfold span
| Mathlib.RingTheory.Ideal.Operations.522_0.5qK551sG47yBciY | theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
r s : R
⊢ span {r} * span {s} = span {r * s} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | unfold span | theorem span_singleton_mul_span_singleton (r s : R) :
span {r} * span {s} = (span {r * s} : Ideal R) := by
| Mathlib.RingTheory.Ideal.Operations.527_0.5qK551sG47yBciY | theorem span_singleton_mul_span_singleton (r s : R) :
span {r} * span {s} = (span {r * s} : Ideal R) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
r s : R
⊢ Submodule.span R {r} * Submodule.span R {s} = Submodule.span R {r * s} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Submodule.span_mul_span, Set.singleton_mul_singleton] | theorem span_singleton_mul_span_singleton (r s : R) :
span {r} * span {s} = (span {r * s} : Ideal R) := by
unfold span
| Mathlib.RingTheory.Ideal.Operations.527_0.5qK551sG47yBciY | theorem span_singleton_mul_span_singleton (r s : R) :
span {r} * span {s} = (span {r * s} : Ideal R) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : R
n : ℕ
⊢ span {s} ^ n = span {s ^ n} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | induction' n with n ih | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by
| Mathlib.RingTheory.Ideal.Operations.533_0.5qK551sG47yBciY | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) | Mathlib_RingTheory_Ideal_Operations |
case zero
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : R
⊢ span {s} ^ Nat.zero = span {s ^ Nat.zero} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp [Set.singleton_one] | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by
induction' n with n ih; · | Mathlib.RingTheory.Ideal.Operations.533_0.5qK551sG47yBciY | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) | Mathlib_RingTheory_Ideal_Operations |
case succ
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : R
n : ℕ
ih : span {s} ^ n = span {s ^ n}
⊢ span {s} ^ Nat.succ n = span {s ^ Nat.succ n} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [pow_succ, ih, span_singleton_mul_span_singleton] | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by
induction' n with n ih; · simp [Set.singleton_one]
| Mathlib.RingTheory.Ideal.Operations.533_0.5qK551sG47yBciY | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
x y : R
I : Ideal R
⊢ x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [mul_comm, mem_mul_span_singleton] | theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by
| Mathlib.RingTheory.Ideal.Operations.542_0.5qK551sG47yBciY | theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
⊢ (∀ {zI : R}, zI ∈ I → zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [mem_span_singleton_mul] | theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} :
I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI :=
show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by
| Mathlib.RingTheory.Ideal.Operations.546_0.5qK551sG47yBciY | theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} :
I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
⊢ span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
| Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
⊢ (∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J) ↔ ∀ z ∈ I, x * z ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | constructor | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
| Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
⊢ (∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J) → ∀ z ∈ I, x * z ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro h zI hzI | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
constructor
· | Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
h : ∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J
zI : R
hzI : zI ∈ I
⊢ x * zI ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact h x (dvd_refl x) zI hzI | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
constructor
· intro h zI hzI
| Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
⊢ (∀ z ∈ I, x * z ∈ J) → ∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rintro h _ ⟨z, rfl⟩ zI hzI | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
constructor
· intro h zI hzI
exact h x (dvd_refl x) zI hzI
· | Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
case mpr.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
h : ∀ z ∈ I, x * z ∈ J
z zI : R
hzI : zI ∈ I
⊢ x * z * zI ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [mul_comm x z, mul_assoc] | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
constructor
· intro h zI hzI
exact h x (dvd_refl x) zI hzI
· rintro h _ ⟨z, rfl⟩ zI hzI
| Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
case mpr.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
h : ∀ z ∈ I, x * z ∈ J
z zI : R
hzI : zI ∈ I
⊢ z * (x * zI) ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact J.mul_mem_left _ (h zI hzI) | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
constructor
· intro h zI hzI
exact h x (dvd_refl x) zI hzI
· rintro h _ ⟨z, rfl⟩ zI hzI
rw [mul_comm x z, mul_assoc]
| Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x y : R
I J : Ideal R
⊢ span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] | theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} :
span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by
| Mathlib.RingTheory.Ideal.Operations.563_0.5qK551sG47yBciY | theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} :
span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝¹ : CommSemiring R
I J K L : Ideal R
inst✝ : IsDomain R
x : R
hx : x ≠ 0
⊢ span {x} * I ≤ span {x} * J ↔ I ≤ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx,
exists_eq_right', SetLike.le_def] | theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) :
span {x} * I ≤ span {x} * J ↔ I ≤ J := by
| Mathlib.RingTheory.Ideal.Operations.568_0.5qK551sG47yBciY | theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) :
span {x} * I ≤ span {x} * J ↔ I ≤ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝¹ : CommSemiring R
I J K L : Ideal R
inst✝ : IsDomain R
x : R
hx : x ≠ 0
⊢ I * span {x} ≤ J * span {x} ↔ I ≤ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx | theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) :
I * span {x} ≤ J * span {x} ↔ I ≤ J := by
| Mathlib.RingTheory.Ideal.Operations.574_0.5qK551sG47yBciY | theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) :
I * span {x} ≤ J * span {x} ↔ I ≤ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝¹ : CommSemiring R
I J K L : Ideal R
inst✝ : IsDomain R
x : R
hx : x ≠ 0
⊢ span {x} * I = span {x} * J ↔ I = J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] | theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) :
span {x} * I = span {x} * J ↔ I = J := by
| Mathlib.RingTheory.Ideal.Operations.579_0.5qK551sG47yBciY | theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) :
span {x} * I = span {x} * J ↔ I = J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝¹ : CommSemiring R
I J K L : Ideal R
inst✝ : IsDomain R
x : R
hx : x ≠ 0
⊢ I * span {x} = J * span {x} ↔ I = J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] | theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) :
I * span {x} = J * span {x} ↔ I = J := by
| Mathlib.RingTheory.Ideal.Operations.584_0.5qK551sG47yBciY | theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) :
I * span {x} = J * span {x} ↔ I = J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R
⊢ I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] | theorem eq_span_singleton_mul {x : R} (I J : Ideal R) :
I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by
| Mathlib.RingTheory.Ideal.Operations.599_0.5qK551sG47yBciY | theorem eq_span_singleton_mul {x : R} (I J : Ideal R) :
I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x y : R
I J : Ideal R
⊢ span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] | theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) :
span {x} * I = span {y} * J ↔
(∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ :=
by | Mathlib.RingTheory.Ideal.Operations.604_0.5qK551sG47yBciY | theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) :
span {x} * I = span {y} * J ↔
(∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
m : Multiset R
⊢ Multiset.prod (Multiset.map (fun x => span {x}) 0) = span {Multiset.prod 0} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp | @[simp]
theorem multiset_prod_span_singleton (m : Multiset R) :
(m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) :=
Multiset.induction_on m (by | Mathlib.RingTheory.Ideal.Operations.620_0.5qK551sG47yBciY | @[simp]
theorem multiset_prod_span_singleton (m : Multiset R) :
(m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
m✝ : Multiset R
a : R
m : Multiset R
ih : Multiset.prod (Multiset.map (fun x => span {x}) m) = span {Multiset.prod m}
⊢ Multiset.prod (Multiset.map (fun x => span {x}) (a ::ₘ m)) = span {Multiset.prod (a ::ₘ m)} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] | @[simp]
theorem multiset_prod_span_singleton (m : Multiset R) :
(m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) :=
Multiset.induction_on m (by simp) fun a m ih => by
| Mathlib.RingTheory.Ideal.Operations.620_0.5qK551sG47yBciY | @[simp]
theorem multiset_prod_span_singleton (m : Multiset R) :
(m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → R
hI : Set.Pairwise (↑s) (IsCoprime on I)
⊢ (Finset.inf s fun i => span {I i}) = span {∏ i in s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | ext x | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by
| Mathlib.RingTheory.Ideal.Operations.627_0.5qK551sG47yBciY | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} | Mathlib_RingTheory_Ideal_Operations |
case h
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → R
hI : Set.Pairwise (↑s) (IsCoprime on I)
x : R
⊢ (x ∈ Finset.inf s fun i => span {I i}) ↔ x ∈ span {∏ i in s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by
ext x
| Mathlib.RingTheory.Ideal.Operations.627_0.5qK551sG47yBciY | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} | Mathlib_RingTheory_Ideal_Operations |
case h
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → R
hI : Set.Pairwise (↑s) (IsCoprime on I)
x : R
⊢ (∀ i ∈ s, I i ∣ x) ↔ ∏ i in s, I 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by
ext x
simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]
| Mathlib.RingTheory.Ideal.Operations.627_0.5qK551sG47yBciY | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι✝ : Type u_1
inst✝¹ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
inst✝ : Fintype ι
I : ι → R
hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j)
⊢ ⨅ i, span {I i} = span {∏ i : ι, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] | theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R}
(hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) :
⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by
| Mathlib.RingTheory.Ideal.Operations.635_0.5qK551sG47yBciY | theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R}
(hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) :
⨅ i, span ({I i} : Set R) = span {∏ i, I i} | Mathlib_RingTheory_Ideal_Operations |
case hI
R : Type u
ι✝ : Type u_1
inst✝¹ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
inst✝ : Fintype ι
I : ι → R
hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j)
⊢ Set.Pairwise (↑Finset.univ) (IsCoprime on fun i => 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rwa [Finset.coe_univ, Set.pairwise_univ] | theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R}
(hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) :
⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by
rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton]
| Mathlib.RingTheory.Ideal.Operations.635_0.5qK551sG47yBciY | theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R}
(hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) :
⨅ i, span ({I i} : Set R) = span {∏ i, I i} | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι✝ : Type u_1
inst✝² : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Fintype ι
I : ι → ℕ
hI : ∀ (i j : ι), i ≠ j → Nat.Coprime (I i) (I j)
⊢ ⨅ i, span {↑(I i)} = span {↑(∏ i : ι, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [iInf_span_singleton, Nat.cast_prod] | theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι]
{I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) :
⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by
| Mathlib.RingTheory.Ideal.Operations.642_0.5qK551sG47yBciY | theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι]
{I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) :
⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι✝ : Type u_1
inst✝² : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Fintype ι
I : ι → ℕ
hI : ∀ (i j : ι), i ≠ j → Nat.Coprime (I i) (I j)
⊢ ∀ (i j : ι), i ≠ j → IsCoprime ↑(I i) ↑(I j) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact fun i j h ↦ (hI i j h).cast | theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι]
{I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) :
⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by
rw [iInf_span_singleton, Nat.cast_prod]
| Mathlib.RingTheory.Ideal.Operations.642_0.5qK551sG47yBciY | theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι]
{I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) :
⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y : R
⊢ span {x} ⊔ span {y} = ⊤ ↔ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [eq_top_iff_one, Submodule.mem_sup] | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
| Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y : R
⊢ (∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1) ↔ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | constructor | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
| Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
case mp
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y : R
⊢ (∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1) → IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rintro ⟨u, hu, v, hv, h1⟩ | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
constructor
· | Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y u : R
hu : u ∈ span {x}
v : R
hv : v ∈ span {y}
h1 : u + v = 1
⊢ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [mem_span_singleton'] at hu hv | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
constructor
· rintro ⟨u, hu, v, hv, h1⟩
| Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y u : R
hu : ∃ a, a * x = u
v : R
hv : ∃ a, a * y = v
h1 : u + v = 1
⊢ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← hu.choose_spec, ← hv.choose_spec] at h1 | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
constructor
· rintro ⟨u, hu, v, hv, h1⟩
rw [mem_span_singleton'] at hu hv
| Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y u : R
hu : ∃ a, a * x = u
v : R
hv : ∃ a, a * y = v
h1 : Exists.choose hu * x + Exists.choose hv * y = 1
⊢ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact ⟨_, _, h1⟩ | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
constructor
· rintro ⟨u, hu, v, hv, h1⟩
rw [mem_span_singleton'] at hu hv
rw [← hu.choose_spec, ← hv.choose_spec] at h1
| Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
case mpr
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝ : CommSemiring R
x y : R
⊢ IsCoprime x y → ∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact fun ⟨u, v, h1⟩ =>
⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
constructor
· rintro ⟨u, hu, v, hv, h1⟩
rw [mem_span_singleton'] at hu hv
rw [← hu.choose_spec, ← hv.choose_spec] at h1
exact ⟨_,... | Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
⊢ Multiset.prod s ≤ Multiset.inf s | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | classical
refine' s.induction_on _ _
· rw [Multiset.inf_zero]
exact le_top
intro a s ih
rw [Multiset.prod_cons, Multiset.inf_cons]
exact le_trans mul_le_inf (inf_le_inf le_rfl ih) | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
| Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
⊢ Multiset.prod s ≤ Multiset.inf s | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' s.induction_on _ _ | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
classical
| Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
⊢ Multiset.prod 0 ≤ Multiset.inf 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Multiset.inf_zero] | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
classical
refine' s.induction_on _ _
· | Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
⊢ Multiset.prod 0 ≤ ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact le_top | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
classical
refine' s.induction_on _ _
· rw [Multiset.inf_zero]
| Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
⊢ ∀ ⦃a : Ideal R⦄ {s : Multiset (Ideal R)},
Multiset.prod s ≤ Multiset.inf s → Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro a s ih | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
classical
refine' s.induction_on _ _
· rw [Multiset.inf_zero]
exact le_top
| Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s✝ : Multiset (Ideal R)
a : Ideal R
s : Multiset (Ideal R)
ih : Multiset.prod s ≤ Multiset.inf s
⊢ Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Multiset.prod_cons, Multiset.inf_cons] | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
classical
refine' s.induction_on _ _
· rw [Multiset.inf_zero]
exact le_top
intro a s ih
| Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s✝ : Multiset (Ideal R)
a : Ideal R
s : Multiset (Ideal R)
ih : Multiset.prod s ≤ Multiset.inf s
⊢ a * Multiset.prod s ≤ a ⊓ Multiset.inf s | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact le_trans mul_le_inf (inf_le_inf le_rfl ih) | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by
classical
refine' s.induction_on _ _
· rw [Multiset.inf_zero]
exact le_top
intro a s ih
rw [Multiset.prod_cons, Multiset.inf_cons]
| Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ J = ⊤
i : R
hi : i ∈ I ⊔ K
⊢ i ∈ I ⊔ J * K | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [eq_top_iff_one] at h | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K :=
le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by
| Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : 1 ∈ I ⊔ J
i : R
hi : i ∈ I ⊔ K
⊢ i ∈ I ⊔ J * K | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Submodule.mem_sup] at h hi ⊢ | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K :=
le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by
rw [eq_top_iff_one] at h; | Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : ∃ y ∈ I, ∃ z ∈ J, y + z = 1
i : R
hi : ∃ y ∈ I, ∃ z ∈ K, y + z = i
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | obtain ⟨i1, hi1, j, hj, h⟩ := h | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K :=
le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by
rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢
| Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K | Mathlib_RingTheory_Ideal_Operations |
case intro.intro.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
i : R
hi : ∃ y ∈ I, ∃ z ∈ K, y + z = i
i1 : R
hi1 : i1 ∈ I
j : R
hj : j ∈ J
h : i1 + j = 1
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | obtain ⟨i', hi', k, hk, hi⟩ := hi | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K :=
le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by
rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢
obtain ⟨i1, hi1, j, hj, h⟩ := h; | Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K | Mathlib_RingTheory_Ideal_Operations |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
i i1 : R
hi1 : i1 ∈ I
j : R
hj : j ∈ J
h : i1 + j = 1
i' : R
hi' : i' ∈ I
k : R
hk : k ∈ K
hi : i' + k = i
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K :=
le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by
rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢
obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi
| Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K | Mathlib_RingTheory_Ideal_Operations |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
i i1 : R
hi1 : i1 ∈ I
j : R
hj : j ∈ J
h : i1 + j = 1
i' : R
hi' : i' ∈ I
k : R
hk : k ∈ K
hi : i' + k = i
⊢ i' + i1 * k + j * k = i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [add_assoc, ← add_mul, h, one_mul, hi] | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K :=
le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by
rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢
obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi
refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1)... | Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ K = ⊤
⊢ I ⊔ J * K = I ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [mul_comm] | theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by
| Mathlib.RingTheory.Ideal.Operations.694_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ K = ⊤
⊢ I ⊔ K * J = I ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact sup_mul_eq_of_coprime_left h | theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by
rw [mul_comm]
| Mathlib.RingTheory.Ideal.Operations.694_0.5qK551sG47yBciY | theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ J = ⊤
⊢ I * K ⊔ J = K ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [sup_comm] at h | theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by
| Mathlib.RingTheory.Ideal.Operations.699_0.5qK551sG47yBciY | theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : J ⊔ I = ⊤
⊢ I * K ⊔ J = K ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] | theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by
rw [sup_comm] at h
| Mathlib.RingTheory.Ideal.Operations.699_0.5qK551sG47yBciY | theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : K ⊔ J = ⊤
⊢ I * K ⊔ J = I ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [sup_comm] at h | theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by
| Mathlib.RingTheory.Ideal.Operations.704_0.5qK551sG47yBciY | theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : J ⊔ K = ⊤
⊢ I * K ⊔ J = I ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] | theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by
rw [sup_comm] at h
| Mathlib.RingTheory.Ideal.Operations.704_0.5qK551sG47yBciY | theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
s : Finset ι
J : ι → Ideal R
h : ∀ i ∈ s, I ⊔ J i = ⊤
⊢ (fun J => I ⊔ J = ⊤) 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp_rw [one_eq_top, sup_top_eq] | theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
(I ⊔ ∏ i in s, J i) = ⊤ :=
Finset.prod_induction _ (fun J => I ⊔ J = ⊤)
(fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK)
(by | Mathlib.RingTheory.Ideal.Operations.709_0.5qK551sG47yBciY | theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
(I ⊔ ∏ i in s, J i) = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ I ⊔ J ^ n = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← Finset.card_range n, ← Finset.prod_const] | theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by
| Mathlib.RingTheory.Ideal.Operations.733_0.5qK551sG47yBciY | theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ I ⊔ ∏ _x in Finset.range n, J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact sup_prod_eq_top fun _ _ => h | theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by
rw [← Finset.card_range n, ← Finset.prod_const]
| Mathlib.RingTheory.Ideal.Operations.733_0.5qK551sG47yBciY | theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ I ^ n ⊔ J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← Finset.card_range n, ← Finset.prod_const] | theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by
| Mathlib.RingTheory.Ideal.Operations.738_0.5qK551sG47yBciY | theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ (∏ _x in Finset.range n, I) ⊔ J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact prod_sup_eq_top fun _ _ => h | theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by
rw [← Finset.card_range n, ← Finset.prod_const]
| Mathlib.RingTheory.Ideal.Operations.738_0.5qK551sG47yBciY | theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ I * ⊥ = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp | theorem mul_bot : I * ⊥ = ⊥ := by | Mathlib.RingTheory.Ideal.Operations.750_0.5qK551sG47yBciY | theorem mul_bot : I * ⊥ = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ ⊥ * I = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp | theorem bot_mul : ⊥ * I = ⊥ := by | Mathlib.RingTheory.Ideal.Operations.754_0.5qK551sG47yBciY | theorem bot_mul : ⊥ * I = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
m n : ℕ
h : m ≤ n
⊢ I ^ n ≤ I ^ m | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | cases' Nat.exists_eq_add_of_le h with k hk | theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by
| Mathlib.RingTheory.Ideal.Operations.793_0.5qK551sG47yBciY | theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m | Mathlib_RingTheory_Ideal_Operations |
case intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
m n : ℕ
h : m ≤ n
k : ℕ
hk : n = m + k
⊢ I ^ n ≤ I ^ m | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [hk, pow_add] | theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by
cases' Nat.exists_eq_add_of_le h with k hk
| Mathlib.RingTheory.Ideal.Operations.793_0.5qK551sG47yBciY | theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m | Mathlib_RingTheory_Ideal_Operations |
case intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
m n : ℕ
h : m ≤ n
k : ℕ
hk : n = m + k
⊢ I ^ m * I ^ k ≤ I ^ m | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact le_trans mul_le_inf inf_le_left | theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by
cases' Nat.exists_eq_add_of_le h with k hk
rw [hk, pow_add]
| Mathlib.RingTheory.Ideal.Operations.793_0.5qK551sG47yBciY | theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L I J : Ideal R
e : I ≤ J
n : ℕ
⊢ I ^ n ≤ J ^ n | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | induction' n with _ hn | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by
| Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n | Mathlib_RingTheory_Ideal_Operations |
case zero
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L I J : Ideal R
e : I ≤ J
⊢ I ^ Nat.zero ≤ J ^ Nat.zero | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [pow_zero, pow_zero] | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by
induction' n with _ hn
· | Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n | Mathlib_RingTheory_Ideal_Operations |
case succ
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L I J : Ideal R
e : I ≤ J
n✝ : ℕ
hn : I ^ n✝ ≤ J ^ n✝
⊢ I ^ Nat.succ n✝ ≤ J ^ Nat.succ n✝ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [pow_succ, pow_succ] | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by
induction' n with _ hn
· rw [pow_zero, pow_zero]
· | Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n | Mathlib_RingTheory_Ideal_Operations |
case succ
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L I J : Ideal R
e : I ≤ J
n✝ : ℕ
hn : I ^ n✝ ≤ J ^ n✝
⊢ I * I ^ n✝ ≤ J * J ^ n✝ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact Ideal.mul_mono e hn | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by
induction' n with _ hn
· rw [pow_zero, pow_zero]
· rw [pow_succ, pow_succ]
| Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY | theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝² : CommSemiring R✝
I✝ J✝ K L : Ideal R✝
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
I J : Ideal R
h : I = ⊥ ∨ J = ⊥
⊢ I * J = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | cases' h with h h | theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ :=
⟨fun hij =>
or_iff_not_imp_left.mpr fun I_ne_bot =>
J.eq_bot_iff.mpr fun j hj =>
let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot
Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_... | Mathlib.RingTheory.Ideal.Operations.812_0.5qK551sG47yBciY | theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ | Mathlib_RingTheory_Ideal_Operations |
case inl
R✝ : Type u
ι : Type u_1
inst✝² : CommSemiring R✝
I✝ J✝ K L : Ideal R✝
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
I J : Ideal R
h : I = ⊥
⊢ I * J = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← Ideal.mul_bot, h, Ideal.mul_comm] | theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ :=
⟨fun hij =>
or_iff_not_imp_left.mpr fun I_ne_bot =>
J.eq_bot_iff.mpr fun j hj =>
let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot
Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_... | Mathlib.RingTheory.Ideal.Operations.812_0.5qK551sG47yBciY | theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ | Mathlib_RingTheory_Ideal_Operations |
case inr
R✝ : Type u
ι : Type u_1
inst✝² : CommSemiring R✝
I✝ J✝ K L : Ideal R✝
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
I J : Ideal R
h : J = ⊥
⊢ I * J = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← Ideal.mul_bot, h, Ideal.mul_comm] | theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ :=
⟨fun hij =>
or_iff_not_imp_left.mpr fun I_ne_bot =>
J.eq_bot_iff.mpr fun j hj =>
let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot
Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_... | Mathlib.RingTheory.Ideal.Operations.812_0.5qK551sG47yBciY | theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝² : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝¹ : CommRing R
inst✝ : IsDomain R
s : Multiset (Ideal R)
⊢ Multiset.prod s = ⊥ ↔ ∃ I ∈ s, I = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [bot_eq_zero, prod_zero_iff_exists_zero] | /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/
theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} :
s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by
| Mathlib.RingTheory.Ideal.Operations.825_0.5qK551sG47yBciY | /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/
theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} :
s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝² : CommSemiring R✝
I J K L : Ideal R✝
R : Type u_2
inst✝¹ : CommRing R
inst✝ : IsDomain R
s : Multiset (Ideal R)
⊢ (∃ r, ∃ (_ : r ∈ s), r = 0) ↔ ∃ I ∈ s, I = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp | /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/
theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} :
s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by
rw [bot_eq_zero, prod_zero_iff_exists_zero]
| Mathlib.RingTheory.Ideal.Operations.825_0.5qK551sG47yBciY | /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/
theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} :
s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
w x y z : R
⊢ span {w, x} * span {y, z} = span {w * y, w * z, x * y, x * z} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] | theorem span_pair_mul_span_pair (w x y z : R) :
(span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by
| Mathlib.RingTheory.Ideal.Operations.832_0.5qK551sG47yBciY | theorem span_pair_mul_span_pair (w x y z : R) :
(span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ Codisjoint I J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [IsCoprime, codisjoint_iff] | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
| Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ (∃ a b, a * I + b * J = 1) ↔ I ⊔ J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | constructor | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
| Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ (∃ a b, a * I + b * J = 1) → I ⊔ J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rintro ⟨x, y, hxy⟩ | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· | Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ I ⊔ J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [eq_top_iff_one] | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
| Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ 1 ∈ I ⊔ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
| Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro.a
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ 1 ∈ x * I + y * J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [hxy] | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
| Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mp.intro.intro.a
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [one_eq_top, Submodule.mem_top] | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
rw [hxy]
| Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ I ⊔ J = ⊤ → ∃ a b, a * I + b * J = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro h | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
rw [hxy]
simp only [one_... | Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ J = ⊤
⊢ ∃ a b, a * I + b * J = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' ⟨1, 1, _⟩ | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
rw [hxy]
simp only [one_... | Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ J = ⊤
⊢ 1 * I + 1 * J = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
rw [hxy]
simp only [one_... | Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ I + J = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] | theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by
| Mathlib.RingTheory.Ideal.Operations.850_0.5qK551sG47yBciY | theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← add_eq_one_iff, isCoprime_iff_add] | theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by
| Mathlib.RingTheory.Ideal.Operations.853_0.5qK551sG47yBciY | theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ I ⊔ J = ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [isCoprime_iff_codisjoint, codisjoint_iff] | theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by
| Mathlib.RingTheory.Ideal.Operations.856_0.5qK551sG47yBciY | theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists,
← isCoprime_iff_sup_eq] | open List in
theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by
| Mathlib.RingTheory.Ideal.Operations.859_0.5qK551sG47yBciY | open List in
theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ TFAE [IsCoprime I J, IsCoprime I J, IsCoprime I J, IsCoprime I J, IsCoprime I J] | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp | open List in
theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by
rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists,
← isCoprime_iff_sup_eq]
| Mathlib.RingTheory.Ideal.Operations.859_0.5qK551sG47yBciY | open List in
theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ IsCoprime (span {x}) (span {y}) ↔ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
mem_span_singleton] | theorem isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
| Mathlib.RingTheory.Ideal.Operations.876_0.5qK551sG47yBciY | theorem isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ (∃ a b, y ∣ b ∧ a * x + b = 1) ↔ IsCoprime 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | constructor | theorem isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
mem_span_singleton]
| Mathlib.RingTheory.Ideal.Operations.876_0.5qK551sG47yBciY | theorem isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y | Mathlib_RingTheory_Ideal_Operations |
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