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 mp
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... | rintro ⟨a, _, ⟨b, rfl⟩, e⟩ | 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]
constructor
· | 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 |
case mp.intro.intro.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y a b : R
e : a * x + y * 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... | exact ⟨a, b, mul_comm b y ▸ e⟩ | 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]
constructor
· rintro ⟨a, _, ⟨b, rfl⟩, e⟩; | 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 |
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ IsCoprime x y → ∃ a b, y ∣ b ∧ a * x + 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
-/
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 ⟨a, b, e⟩ | 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]
constructor
· rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ ... | 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 |
case mpr.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y a b : R
e : a * x + b * y = 1
⊢ ∃ a b, y ∣ b ∧ a * x + 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
-/
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 ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ | 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]
constructor
· rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ ... | 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
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, IsCoprime I (J j)
⊢ IsCoprime I (⨅ j ∈ s, J 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... | classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
set K := ⨅ j ∈ s, J j
calc
1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_inse... | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, IsCoprime I (J j)
⊢ IsCoprime I (⨅ j ∈ s, J 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 [isCoprime_iff_add] at * | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, I + J j = 1
⊢ I + ⨅ j ∈ s, J 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... | induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
set K := ⨅ j ∈ s, J j
calc
1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
_ = I + K*(I + J i)... | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, I + J j = 1
⊢ I + ⨅ j ∈ s, J 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... | induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
set K := ⨅ j ∈ s, J j
calc
1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
_ = I + K*(I + J i)... | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
case empty
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
hf : ∀ j ∈ ∅, I + J j = 1
⊢ I + ⨅ j ∈ ∅, J 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... | | empty =>
simp | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
case empty
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
hf : ∀ j ∈ ∅, I + J j = 1
⊢ I + ⨅ j ∈ ∅, J 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 | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1
hf : ∀ j ∈ insert i s, I + J j = 1
⊢ I + ⨅ j ∈ insert i s, J 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... | | @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
set K := ⨅ j ∈ s, J j
calc
1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
_ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one]
_ ... | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1
hf : ∀ j ∈ insert i s, I + J j = 1
⊢ I + ⨅ j ∈ insert i s, J 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 [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
| Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1
hf : ∀ j ∈ insert i s, I + J j = 1
⊢ 1 ≤ I + (⨅ x ∈ s, J x) ⊓ 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | set K := ⨅ j ∈ s, J j | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, e... | Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K✝ L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hf : ∀ j ∈ insert i s, I + J j = 1
K : Ideal R := ⨅ j ∈ s, J j
hs : (∀ j ∈ s, I + J j = 1) → I + K = 1
⊢ 1 ≤ I + K ⊓ 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | calc
1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
_ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one]
_ = (1+K)*I + K*J i := by ring
_ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, e... | Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K✝ L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hf : ∀ j ∈ insert i s, I + J j = 1
K : Ideal R := ⨅ j ∈ s, J j
hs : (∀ j ∈ s, I + J j = 1) → I + K = 1
⊢ I + K = I + K * (I + 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
-/
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 [hf i (Finset.mem_insert_self i s), mul_one] | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, e... | Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K✝ L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hf : ∀ j ∈ insert i s, I + J j = 1
K : Ideal R := ⨅ j ∈ s, J j
hs : (∀ j ∈ s, I + J j = 1) → I + K = 1
⊢ I + K * (I + J i) = (1 + K) * I + K * 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | ring | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| @insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, e... | Mathlib.RingTheory.Ideal.Operations.884_0.5qK551sG47yBciY | theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r s : R
⊢ s ∈
{
toAddSubsemigroup :=
{ carrier := {r | ∃ n, r ^ n ∈ I},
add_mem' :=
(_ :
∀ {x y : R}, x ∈ {r | ∃ n, r ^ n ∈ I} → y ∈ {r | ∃ n, r ^ n ∈ I} → x + y ∈ {r... | /-
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 ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ | /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/
def radical (I : Ideal R) : Ideal R where
carrier := { r | ∃ n : ℕ, r ^ n ∈ I }
zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩
add_mem' :=
fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ =>
⟨m + n,
(add_pow x y (m... | Mathlib.RingTheory.Ideal.Operations.900_0.5qK551sG47yBciY | /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/
def radical (I : Ideal R) : Ideal R where
carrier | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ radical I = I ↔ IsRadical 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 [le_antisymm_iff, and_iff_left le_radical, IsRadical] | /-- An ideal is radical iff it is equal to its radical. -/
theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by
| Mathlib.RingTheory.Ideal.Operations.932_0.5qK551sG47yBciY | /-- An ideal is radical iff it is equal to its radical. -/
theorem radical_eq_iff : I.radical = I ↔ I.IsRadical | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ radical (I * J) = radical I ⊓ radical 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... | refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ | theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
| Mathlib.RingTheory.Ideal.Operations.1002_0.5qK551sG47yBciY | theorem radical_mul : radical (I * J) = radical I ⊓ radical J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ radical (I * J) ≤ radical I ⊓ radical 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... | have := radical_mono <| @mul_le_inf _ _ I J | theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
| Mathlib.RingTheory.Ideal.Operations.1002_0.5qK551sG47yBciY | theorem radical_mul : radical (I * J) = radical I ⊓ radical J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
this : radical (I * J) ≤ radical (I ⊓ J)
⊢ radical (I * J) ≤ radical I ⊓ radical 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 [radical_inf I J] at this | theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
have := radical_mono <| @mul_le_inf _ _ I J
| Mathlib.RingTheory.Ideal.Operations.1002_0.5qK551sG47yBciY | theorem radical_mul : radical (I * J) = radical I ⊓ radical J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
this : radical (I * J) ≤ radical I ⊓ radical J
⊢ radical (I * J) ≤ radical I ⊓ radical 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... | assumption | theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
have := radical_mono <| @mul_le_inf _ _ I J
simp_rw [radical_inf I J] at this
| Mathlib.RingTheory.Ideal.Operations.1002_0.5qK551sG47yBciY | theorem radical_mul : radical (I * J) = radical I ⊓ radical J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
m : Ideal R
hrm : r ∉ radical m
him : I ≤ m
hm : ∀ z ∈ {K | r ∉ radical K}, m ≤ z → z = m
this : ∀ x ∉ m, r ∈ radical (m ⊔ span {x})
⊢ 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... | rintro rfl | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
by_contradiction fun hri ↦
let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
... | Mathlib.RingTheory.Ideal.Operations.1016_0.5qK551sG47yBciY | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
hrm : r ∉ radical ⊤
him : I ≤ ⊤
hm : ∀ z ∈ {K | r ∉ radical K}, ⊤ ≤ z → z = ⊤
this : ∀ x ∉ ⊤, r ∈ radical (⊤ ⊔ span {x})
⊢ False | /-
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 [radical_top] at hrm | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
by_contradiction fun hri ↦
let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
... | Mathlib.RingTheory.Ideal.Operations.1016_0.5qK551sG47yBciY | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
hrm : r ∉ ⊤
him : I ≤ ⊤
hm : ∀ z ∈ {K | r ∉ radical K}, ⊤ ≤ z → z = ⊤
this : ∀ x ∉ ⊤, r ∈ radical (⊤ ⊔ span {x})
⊢ False | /-
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 hrm trivial | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
by_contradiction fun hri ↦
let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
... | Mathlib.RingTheory.Ideal.Operations.1016_0.5qK551sG47yBciY | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
m : Ideal R
hrm : r ∉ radical m
him : I ≤ m
hm : ∀ z ∈ {K | r ∉ radical K}, m ≤ z → z = m
this : ∀ x ∉ m, r ∈ radical (m ⊔ span {x})
x y : R
hxym : x * y ∈ m
hxm : x ∉ m
hym : y ∉ m
n : 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_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x),
mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc] | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
by_contradiction fun hri ↦
let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
... | Mathlib.RingTheory.Ideal.Operations.1016_0.5qK551sG47yBciY | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
m : Ideal R
hrm : r ∉ radical m
him : I ≤ m
hm : ∀ z ∈ {K | r ∉ radical K}, m ≤ z → z = m
this : ∀ x ∉ m, r ∈ radical (m ⊔ span {x})
x y : R
hxym : x * y ∈ m
hxm : x ∉ m
hym : y ∉ m
n : 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... | refine'
m.add_mem (m.mul_mem_right _ hpm)
(m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym)) | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
by_contradiction fun hri ↦
let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
... | Mathlib.RingTheory.Ideal.Operations.1016_0.5qK551sG47yBciY | theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n✝ n : ℕ
ih : ⊤ ^ n = ⊤
⊢ ⊤ ^ 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, ih, top_mul] | theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ :=
Nat.recOn n one_eq_top fun n ih => by | Mathlib.RingTheory.Ideal.Operations.1068_0.5qK551sG47yBciY | theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
H : n > 0
⊢ ¬Nat.zero > 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... | decide | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I :=
Nat.recOn n (Not.elim (by | Mathlib.RingTheory.Ideal.Operations.1076_0.5qK551sG47yBciY | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n✝ : ℕ
H✝¹ : n✝ > 0
n : ℕ
ih : n > 0 → radical (I ^ n) = radical I
H✝ : Nat.succ n > 0
H : 0 < n
⊢ radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ 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] | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I :=
Nat.recOn n (Not.elim (by decide))
(fun n ih H =>
Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H)
(fun H =>
calc
radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by
| Mathlib.RingTheory.Ideal.Operations.1076_0.5qK551sG47yBciY | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n✝ : ℕ
H✝¹ : n✝ > 0
n : ℕ
ih : n > 0 → radical (I ^ n) = radical I
H✝ : Nat.succ n > 0
H : 0 < n
⊢ radical (I * I ^ n) = radical I ⊓ radical (I ^ 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 radical_mul _ _ | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I :=
Nat.recOn n (Not.elim (by decide))
(fun n ih H =>
Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H)
(fun H =>
calc
radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by
rw [pow_succ]
... | Mathlib.RingTheory.Ideal.Operations.1076_0.5qK551sG47yBciY | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n✝ : ℕ
H✝¹ : n✝ > 0
n : ℕ
ih : n > 0 → radical (I ^ n) = radical I
H✝ : Nat.succ n > 0
H : 0 < n
⊢ radical I ⊓ radical (I ^ n) = radical I ⊓ radical 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 [ih H] | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I :=
Nat.recOn n (Not.elim (by decide))
(fun n ih H =>
Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H)
(fun H =>
calc
radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by
rw [pow_succ]
... | Mathlib.RingTheory.Ideal.Operations.1076_0.5qK551sG47yBciY | theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L I J P : Ideal R
hp : IsPrime P
⊢ I * J ≤ P ↔ I ≤ P ∨ J ≤ P | /-
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 [or_comm, Ideal.mul_le] | theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by
| Mathlib.RingTheory.Ideal.Operations.1092_0.5qK551sG47yBciY | theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L I J P : Ideal R
hp : IsPrime P
⊢ (∀ r ∈ I, ∀ s ∈ J, r * s ∈ P) ↔ J ≤ P ∨ I ≤ P | /-
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 [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] | theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by
rw [or_comm, Ideal.mul_le]
| Mathlib.RingTheory.Ideal.Operations.1092_0.5qK551sG47yBciY | theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
P : Ideal R
hp : IsPrime P
⊢ Multiset.prod 0 ≤ P ↔ ∃ I ∈ 0, I ≤ P | /-
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 [hp.ne_top] | theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) :
s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P :=
s.induction_on (by | Mathlib.RingTheory.Ideal.Operations.1101_0.5qK551sG47yBciY | theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) :
s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
s✝ : Multiset (Ideal R)
P : Ideal R
hp : IsPrime P
I : Ideal R
s : Multiset (Ideal R)
ih : Multiset.prod s ≤ P ↔ ∃ I ∈ s, I ≤ P
⊢ Multiset.prod (I ::ₘ s) ≤ P ↔ ∃ I_1 ∈ I ::ₘ s, I_1 ≤ P | /-
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 [hp.mul_le, ih] | theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) :
s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P :=
s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by | Mathlib.RingTheory.Ideal.Operations.1101_0.5qK551sG47yBciY | theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) :
s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset ι
f : ι → Ideal R
P : Ideal R
hp : IsPrime P
⊢ Multiset.prod (Multiset.map f s) ≤ P ↔ ∃ i ∈ s, f i ≤ P | /-
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 [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] | theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R}
(hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
| Mathlib.RingTheory.Ideal.Operations.1106_0.5qK551sG47yBciY | theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R}
(hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
I : Ideal R
⊢ ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
⟨this, fun h =>
Or.casesOn h
(fun h =>
Set.Subset.trans h <|
Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
fun h =>
Or.casesOn ... | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
| Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
I : Ideal R
this : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
h✝ : I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
h : I ≤ f b ∨ ∃... | /-
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' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
I : Ideal R
this : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
h✝ : I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
h : I ≤ f b ∨ ∃... | /-
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 Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
I : Ideal R
⊢ ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | generalize hn : s.card = n | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
I : Ideal R
n : ℕ
hn : Finset.card s = n
⊢ ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
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 subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
I : Ideal R
n : ℕ
hn : Finset.card s = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
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 generalizing a b s | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case zero
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
s : Finset ι
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
hn : Finset.card s = Nat.zero
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | clear hp | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case zero
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
s : Finset ι
a b : ι
hn : Finset.card s = Nat.zero
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
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_eq_zero] at hn | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case zero
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
s : Finset ι
a b : ι
hn : s = ∅
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | subst hn | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case zero
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
a b : ι
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑∅, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ ∅, 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
-/
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.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case zero
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
a b : ι
h : I ≤ f a ∨ I ≤ f b
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ ∅, 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
-/
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 [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case succ
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
s : F... | /-
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
replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
Finset.card_eq_succ.1 hn
rcases hn with ⟨i, t, hit, rfl, hn⟩
replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
by_cases Ht : ∃ j ∈ t, f j ≤ f i
· obtain ⟨j, hjt, hfji⟩ : ... | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case succ
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
s : F... | /-
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... | replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
Finset.card_eq_succ.1 hn | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case succ
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
s : F... | /-
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... | rcases hn with ⟨i, t, hit, rfl, hn⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case succ.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ... | /-
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... | replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case succ.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ... | /-
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... | by_cases Ht : ∃ j ∈ t, f j ≤ f i | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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 ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ 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... | obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨... | /-
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... | have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by
rw [Finset.forall_mem_insert] at hp ⊢
exact ⟨hp.1, hp.2.2⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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.forall_mem_insert] at hp ⊢ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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 ⟨hp.1, hp.2.2⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨... | /-
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... | have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨... | /-
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... | have hn' : (insert i u).card = n := by
rwa [Finset.card_insert_of_not_mem] at hn ⊢
exacts [hiu, hju] | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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.card_insert_of_not_mem] at hn ⊢ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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... | exacts [hiu, hju] | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨... | /-
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... | have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by
rw [Finset.coe_insert] at h ⊢
rw [Finset.coe_insert] at h
simp only [Set.biUnion_insert] at h ⊢
rw [← Set.union_assoc (f i : Set R)] at h
erw [Set.union_eq_self_of_subset_right hfji] at h
exact h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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.coe_insert] at h ⊢ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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.coe_insert] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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 [Set.biUnion_insert] at h ⊢ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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 [← Set.union_assoc (f i : Set R)] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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... | erw [Set.union_eq_self_of_subset_right hfji] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfj... | /-
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 | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨... | /-
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... | specialize ih hp' hn' h' | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : Finset.card (insert j u) = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert ... | /-
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' ih.imp id (Or.imp id (Exists.imp fun k => _)) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : Finset.card (insert j u) = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert ... | /-
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 And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case neg
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | by_cases Ha : f a ≤ f i | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by
rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc,
Set.union_right_comm (f a : Set R)] at h
erw [Set.union_eq_self_of_subset_left Ha] at h
exact h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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.coe_insert, Set.biUnion_insert, ← Set.union_assoc,
Set.union_right_comm (f a : Set R)] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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... | erw [Set.union_eq_self_of_subset_left Ha] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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 | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | specialize ih hp.2 hn h' | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬∃ j ∈ t, f... | /-
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... | right | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.h
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬∃ j ∈ 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... | rcases ih with (ih | ih | ⟨k, hkt, ih⟩) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.h.inl
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬∃ 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 Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.h.inr.inl
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬... | /-
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 Or.inl ih | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.h.inr.inr.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime ... | /-
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 Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case neg
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | by_cases Hb : f b ≤ f i | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by
rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc,
Set.union_assoc (f a : Set R)] at h
erw [Set.union_eq_self_of_subset_left Hb] at h
exact h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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.coe_insert, Set.biUnion_insert, ← Set.union_assoc,
Set.union_assoc (f a : Set R)] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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... | erw [Set.union_eq_self_of_subset_left Hb] at h | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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 | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | specialize ih hp.2 hn h' | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬∃ j ∈ t, f... | /-
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... | rcases ih with (ih | ih | ⟨k, hkt, ih⟩) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.inl
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬∃ 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 Or.inl ih | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.inr.inl
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ¬∃ ... | /-
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 Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos.inr.inr.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f... | /-
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 Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case neg
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | by_cases Hi : I ≤ f i | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case neg
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by
simp only [hp.1.inf_le, hp.1.inf_le', not_or]
exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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 [hp.1.inf_le, hp.1.inf_le', not_or] | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : F... | /-
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 ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case neg
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b 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... | rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
case neg.intro.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, IsPrime (f i)) →
Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f 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
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤... | Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | Mathlib_RingTheory_Ideal_Operations |
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