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 intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m ≥ n... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine le_antisymm (iSup_le fun l ↦ ?_) (le_iSup (fun i ↦ LinearMap.ker (f ^ i)) m) | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) := by
obtain ⟨n, hn : ∀ m, n ≤ m → ker (f ^ n) = ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ ?_⟩
| Mathlib.RingTheory.Noetherian.446_0.5UPGNrmhtW81IjE | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) | Mathlib_RingTheory_Noetherian |
case intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m ≥ n... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rcases le_or_lt m l with h | h | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) := by
obtain ⟨n, hn : ∀ m, n ≤ m → ker (f ^ n) = ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ ?_⟩
r... | Mathlib.RingTheory.Noetherian.446_0.5UPGNrmhtW81IjE | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) | Mathlib_RingTheory_Noetherian |
case intro.inl
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [← hn _ (hm.trans h), hn _ hm] | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) := by
obtain ⟨n, hn : ∀ m, n ≤ m → ker (f ^ n) = ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ ?_⟩
r... | Mathlib.RingTheory.Noetherian.446_0.5UPGNrmhtW81IjE | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) | Mathlib_RingTheory_Noetherian |
case intro.inr
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact f.iterateKer.monotone h.le | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) := by
obtain ⟨n, hn : ∀ m, n ≤ m → ker (f ^ n) = ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ ?_⟩
r... | Mathlib.RingTheory.Noetherian.446_0.5UPGNrmhtW81IjE | lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
s : Surjective ⇑f
⊢ Injective ⇑f | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | obtain ⟨n, hn⟩ := eventually_atTop.mp f.eventually_disjoint_ker_pow_range_pow | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f := by
| Mathlib.RingTheory.Noetherian.456_0.5UPGNrmhtW81IjE | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f | Mathlib_RingTheory_Noetherian |
case intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
s : Surjective ⇑f
n : ℕ
hn : ∀ b ≥ n, Disjoint (LinearMap.ker (f ^ b)) (... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | specialize hn (n + 1) (n.le_add_right 1) | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f := by
obtain ⟨n, hn⟩ := eventually_atTop.mp f.eventually_disjoint_ker_pow_range_pow
| Mathlib.RingTheory.Noetherian.456_0.5UPGNrmhtW81IjE | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f | Mathlib_RingTheory_Noetherian |
case intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
s : Surjective ⇑f
n : ℕ
hn : Disjoint (LinearMap.ker (f ^ (n + 1))) (Lin... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [disjoint_iff, LinearMap.range_eq_top.mpr (LinearMap.iterate_surjective s _), inf_top_eq,
LinearMap.ker_eq_bot] at hn | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f := by
obtain ⟨n, hn⟩ := eventually_atTop.mp f.eventually_disjoint_ker_pow_range_pow
specialize hn (n + 1) (n.le_add_right 1)
| Mathlib.RingTheory.Noetherian.456_0.5UPGNrmhtW81IjE | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f | Mathlib_RingTheory_Noetherian |
case intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
s : Surjective ⇑f
n : ℕ
hn : Injective ⇑(f ^ (n + 1))
⊢ Injective ⇑f | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact LinearMap.injective_of_iterate_injective n.succ_ne_zero hn | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f := by
obtain ⟨n, hn⟩ := eventually_atTop.mp f.eventually_disjoint_ker_pow_range_pow
specialize hn (n + 1) (n.le_add_right 1)
rw [di... | Mathlib.RingTheory.Noetherian.456_0.5UPGNrmhtW81IjE | /-- Any surjective endomorphism of a Noetherian module is injective. -/
theorem IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
⊢ ∃ n, ∀ (m : ℕ), n... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | suffices t : ∃ n : ℕ, ∀ m, n ≤ m → f (m + 1) = ⊥ | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
t : ∃ n, ∀ (m : ℕ),... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | obtain ⟨n, w⟩ := t | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
n : ℕ
w ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | use n + 1 | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case h
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
n : ℕ
w : ∀ ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rintro (_ | m) p | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case h.zero
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
n : ℕ
w... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | cases p | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case h.succ
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
n : ℕ
w... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | apply w | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case h.succ.a
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
n : ℕ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact Nat.succ_le_succ_iff.mp p | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case t
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
⊢ ∃ n, ∀ (m ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | obtain ⟨n, w⟩ := monotone_stabilizes_iff_noetherian.mpr inferInstance (partialSups f) | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
case t.intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))
n : ℕ
... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact
⟨n, fun m p =>
(h m).eq_bot_of_ge <| sup_eq_left.1 <| (w (m + 1) <| le_add_right p).symm.trans <| w m p⟩ | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ := by... | Mathlib.RingTheory.Noetherian.472_0.5UPGNrmhtW81IjE | /-- A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero.
-/
theorem IsNoetherian.disjoint_partialSups_eventually_bot
(f : ℕ → Submodule R M) (h : ∀ n, Disjoint (partialSups f n) (f (n + 1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
⊢ N ≃ₗ[R] PUnit.{w + 1} | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | apply Nonempty.some | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
| Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
case h
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
⊢ Nonempty (N ≃ₗ[R] PUnit.{w + 1}) | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | obtain ⟨n, w⟩ :=
IsNoetherian.disjoint_partialSups_eventually_bot (f.tailing i) (f.tailings_disjoint_tailing i) | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
apply Nonempty.some
| Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
case h.intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
n : ℕ
w : ∀ (m : ℕ), n ≤ m → LinearMap.tailing f ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | specialize w n (le_refl n) | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
apply Nonempty.some
obtain ⟨n, w⟩ :=
IsNoetherian.disjoint_partialSups_eventually_bot (f.tailing i... | Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
case h.intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
n : ℕ
w : LinearMap.tailing f i n = ⊥
⊢ Nonempty ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | apply Nonempty.intro | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
apply Nonempty.some
obtain ⟨n, w⟩ :=
IsNoetherian.disjoint_partialSups_eventually_bot (f.tailing i... | Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
case h.intro.val
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
n : ℕ
w : LinearMap.tailing f i n = ⊥
⊢ N ≃ₗ[... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine (LinearMap.tailingLinearEquiv f i n).symm ≪≫ₗ ?_ | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
apply Nonempty.some
obtain ⟨n, w⟩ :=
IsNoetherian.disjoint_partialSups_eventually_bot (f.tailing i... | Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
case h.intro.val
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
n : ℕ
w : LinearMap.tailing f i n = ⊥
⊢ ↥(Lin... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [w] | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
apply Nonempty.some
obtain ⟨n, w⟩ :=
IsNoetherian.disjoint_partialSups_eventually_bot (f.tailing i... | Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
case h.intro.val
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective ⇑f
n : ℕ
w : LinearMap.tailing f i n = ⊥
⊢ ↥⊥ ≃ₗ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | apply Submodule.botEquivPUnit | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} := by
apply Nonempty.some
obtain ⟨n, w⟩ :=
IsNoetherian.disjoint_partialSups_eventually_bot (f.tailing i... | Mathlib.RingTheory.Noetherian.493_0.5UPGNrmhtW81IjE | /-- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial.
-/
noncomputable def IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} | Mathlib_RingTheory_Noetherian |
R : Type ?u.338259
M : Type ?u.338262
inst✝³ : Finite M
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
⊢ span R ↑(Finite.toFinset (_ : Set.Finite ↑s)) = s | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [Set.Finite.coe_toFinset, Submodule.span_eq] | instance (priority := 80) isNoetherian_of_finite (R M) [Finite M] [Semiring R] [AddCommMonoid M]
[Module R M] : IsNoetherian R M :=
⟨fun s => ⟨(s : Set M).toFinite.toFinset, by | Mathlib.RingTheory.Noetherian.529_0.5UPGNrmhtW81IjE | instance (priority | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
h : IsNoetherian R M
⊢ IsNoetherian R ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [isNoetherian_iff_wellFounded] at h ⊢ | theorem isNoetherian_of_submodule_of_noetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M]
(N : Submodule R M) (h : IsNoetherian R M) : IsNoetherian R N := by
| Mathlib.RingTheory.Noetherian.542_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_submodule_of_noetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M]
(N : Submodule R M) (h : IsNoetherian R M) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
h : WellFounded fun x x_1 => x > x_1
⊢ WellFounded fun x x_1 => x > x_1 | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact OrderEmbedding.wellFounded (Submodule.MapSubtype.orderEmbedding N).dual h | theorem isNoetherian_of_submodule_of_noetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M]
(N : Submodule R M) (h : IsNoetherian R M) : IsNoetherian R N := by
rw [isNoetherian_iff_wellFounded] at h ⊢
| Mathlib.RingTheory.Noetherian.542_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_submodule_of_noetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M]
(N : Submodule R M) (h : IsNoetherian R M) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type ?u.341864
inst✝² : Ring R
M : Type ?u.341870
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
h : IsNoetherian R M
⊢ IsNoetherian R (M ⧸ N) | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [isNoetherian_iff_wellFounded] at h ⊢ | instance Submodule.Quotient.isNoetherian {R} [Ring R] {M} [AddCommGroup M] [Module R M]
(N : Submodule R M) [h : IsNoetherian R M] : IsNoetherian R (M ⧸ N) := by
| Mathlib.RingTheory.Noetherian.548_0.5UPGNrmhtW81IjE | instance Submodule.Quotient.isNoetherian {R} [Ring R] {M} [AddCommGroup M] [Module R M]
(N : Submodule R M) [h : IsNoetherian R M] : IsNoetherian R (M ⧸ N) | Mathlib_RingTheory_Noetherian |
R : Type ?u.341864
inst✝² : Ring R
M : Type ?u.341870
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
h : WellFounded fun x x_1 => x > x_1
⊢ WellFounded fun x x_1 => x > x_1 | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact OrderEmbedding.wellFounded (Submodule.comapMkQOrderEmbedding N).dual h | instance Submodule.Quotient.isNoetherian {R} [Ring R] {M} [AddCommGroup M] [Module R M]
(N : Submodule R M) [h : IsNoetherian R M] : IsNoetherian R (M ⧸ N) := by
rw [isNoetherian_iff_wellFounded] at h ⊢
| Mathlib.RingTheory.Noetherian.548_0.5UPGNrmhtW81IjE | instance Submodule.Quotient.isNoetherian {R} [Ring R] {M} [AddCommGroup M] [Module R M]
(N : Submodule R M) [h : IsNoetherian R M] : IsNoetherian R (M ⧸ N) | Mathlib_RingTheory_Noetherian |
R : Type u_1
S : Type u_2
M : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid M
inst✝³ : SMul R S
inst✝² : Module S M
inst✝¹ : Module R M
inst✝ : IsScalarTower R S M
h : IsNoetherian R M
⊢ IsNoetherian S M | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [isNoetherian_iff_wellFounded] at h ⊢ | /-- If `M / S / R` is a scalar tower, and `M / R` is Noetherian, then `M / S` is
also noetherian. -/
theorem isNoetherian_of_tower (R) {S M} [Semiring R] [Semiring S] [AddCommMonoid M] [SMul R S]
[Module S M] [Module R M] [IsScalarTower R S M] (h : IsNoetherian R M) : IsNoetherian S M := by
| Mathlib.RingTheory.Noetherian.554_0.5UPGNrmhtW81IjE | /-- If `M / S / R` is a scalar tower, and `M / R` is Noetherian, then `M / S` is
also noetherian. -/
theorem isNoetherian_of_tower (R) {S M} [Semiring R] [Semiring S] [AddCommMonoid M] [SMul R S]
[Module S M] [Module R M] [IsScalarTower R S M] (h : IsNoetherian R M) : IsNoetherian S M | Mathlib_RingTheory_Noetherian |
R : Type u_1
S : Type u_2
M : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid M
inst✝³ : SMul R S
inst✝² : Module S M
inst✝¹ : Module R M
inst✝ : IsScalarTower R S M
h : WellFounded fun x x_1 => x > x_1
⊢ WellFounded fun x x_1 => x > x_1 | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine' (Submodule.restrictScalarsEmbedding R S M).dual.wellFounded h | /-- If `M / S / R` is a scalar tower, and `M / R` is Noetherian, then `M / S` is
also noetherian. -/
theorem isNoetherian_of_tower (R) {S M} [Semiring R] [Semiring S] [AddCommMonoid M] [SMul R S]
[Module S M] [Module R M] [IsScalarTower R S M] (h : IsNoetherian R M) : IsNoetherian S M := by
rw [isNoetherian_iff_w... | Mathlib.RingTheory.Noetherian.554_0.5UPGNrmhtW81IjE | /-- If `M / S / R` is a scalar tower, and `M / R` is Noetherian, then `M / S` is
also noetherian. -/
theorem isNoetherian_of_tower (R) {S M} [Semiring R] [Semiring S] [AddCommMonoid M] [SMul R S]
[Module S M] [Module R M] [IsScalarTower R S M] (h : IsNoetherian R M) : IsNoetherian S M | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
⊢ IsNoetherian R ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | let ⟨s, hs⟩ := hN | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
| Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
⊢ IsNoetherian R ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | haveI := Classical.decEq M | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
| Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this : DecidableEq M
⊢ IsNoetherian R ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | haveI := Classical.decEq R | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
| Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝ : DecidableEq M
this : DecidableEq R
⊢ IsNoetherian R ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
| Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ IsNoetherian R ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine
@isNoetherian_of_surjective
R ((↑s : Set M) → R) N _ _ _ (Pi.module _ _ _) _ ?_ ?_ isNoetherian_pi | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ (↑↑s → R) →ₗ[R] ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | fapply LinearMap.mk | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ AddHom (↑↑s → R) ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | fapply AddHom.mk | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom.toFun
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ (↑↑s → R) → ↥N | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact fun f => ⟨∑ i in s.attach, f i • i.1, N.sum_mem fun c _ => N.smul_mem _ <| this _ c.2⟩ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom.map_add'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ ∀ (x y : ↑↑s → R),
{ val := ∑ i in Finset.attac... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | intro f g | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom.map_add'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ { val := ∑ i in Finset.attach s, (f +... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | apply Subtype.eq | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ↑{ val := ∑ i in Finset.attach s, (... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | change (∑ i in s.attach, (f i + g i) • _) = _ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ∑ i in Finset.attach s, (f i + g i)... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | simp only [add_smul, Finset.sum_add_distrib] | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ∑ x in Finset.attach s, f x • ↑x + ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rfl | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.map_smul'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ ∀ (r : R) (x : ↑↑s → R),
AddHom.toFun
{
... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | intro c f | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.map_smul'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ AddHom.toFun
{
toFun := fun... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | apply Subtype.eq | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ↑(AddHom.toFun
{
toFu... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | change (∑ i in s.attach, (c • f i) • _) = _ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ∑ i in Finset.attach s, (c • f i) • ↑i ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | simp only [smul_eq_mul, mul_smul] | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ∑ x in Finset.attach s, c • f x • ↑x =
... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact Finset.smul_sum.symm | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ LinearMap.range
{
toAddHom :=
{
t... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [LinearMap.range_eq_top] | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ Surjective
⇑{
toAddHom :=
{
toFun :... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rintro ⟨n, hn⟩ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
⊢ ∃ a,
{
toAddHom :=
{
... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | change n ∈ N at hn | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
⊢ ∃ a,
{
toAddHom :=
{
... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [← hs, ← Set.image_id (s : Set M), Finsupp.mem_span_image_iff_total] at hn | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn✝ : n ∈ N
hn : ∃ l ∈ Finsupp.supported R R ↑s, (Finsupp.tota... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rcases hn with ⟨l, hl1, hl2⟩ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine' ⟨fun x => l x, Subtype.ext _⟩ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | change (∑ i in s.attach, l i • (i : M)) = n | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [@Finset.sum_attach M M s _ fun i => l i • i, ← hl2,
Finsupp.total_apply, Finsupp.sum, eq_comm] | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | refine' Finset.sum_subset hl1 fun x _ hx => _ | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported ... | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [Finsupp.not_mem_support_iff.1 hx, zero_smul] | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N := by
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
... | Mathlib.RingTheory.Noetherian.562_0.5UPGNrmhtW81IjE | theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N | Mathlib_RingTheory_Noetherian |
R : Type u_1
inst✝¹ : Ring R
S : Type u_2
inst✝ : Ring S
f : R →+* S
hf : Surjective ⇑f
H : IsNoetherianRing R
⊢ IsNoetherianRing S | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H ⊢ | theorem isNoetherianRing_of_surjective (R) [Ring R] (S) [Ring S] (f : R →+* S)
(hf : Function.Surjective f) [H : IsNoetherianRing R] : IsNoetherianRing S := by
| Mathlib.RingTheory.Noetherian.612_0.5UPGNrmhtW81IjE | theorem isNoetherianRing_of_surjective (R) [Ring R] (S) [Ring S] (f : R →+* S)
(hf : Function.Surjective f) [H : IsNoetherianRing R] : IsNoetherianRing S | Mathlib_RingTheory_Noetherian |
R : Type u_1
inst✝¹ : Ring R
S : Type u_2
inst✝ : Ring S
f : R →+* S
hf : Surjective ⇑f
H : WellFounded fun x x_1 => x > x_1
⊢ WellFounded fun x x_1 => x > x_1 | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact OrderEmbedding.wellFounded (Ideal.orderEmbeddingOfSurjective f hf).dual H | theorem isNoetherianRing_of_surjective (R) [Ring R] (S) [Ring S] (f : R →+* S)
(hf : Function.Surjective f) [H : IsNoetherianRing R] : IsNoetherianRing S := by
rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H ⊢
| Mathlib.RingTheory.Noetherian.612_0.5UPGNrmhtW81IjE | theorem isNoetherianRing_of_surjective (R) [Ring R] (S) [Ring S] (f : R →+* S)
(hf : Function.Surjective f) [H : IsNoetherianRing R] : IsNoetherianRing S | Mathlib_RingTheory_Noetherian |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
⊢ IsNilpotent (nilradical R) | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | obtain ⟨n, hn⟩ := Ideal.exists_radical_pow_le_of_fg (⊥ : Ideal R) (IsNoetherian.noetherian _) | theorem IsNoetherianRing.isNilpotent_nilradical (R : Type*) [CommRing R] [IsNoetherianRing R] :
IsNilpotent (nilradical R) := by
| Mathlib.RingTheory.Noetherian.628_0.5UPGNrmhtW81IjE | theorem IsNoetherianRing.isNilpotent_nilradical (R : Type*) [CommRing R] [IsNoetherianRing R] :
IsNilpotent (nilradical R) | Mathlib_RingTheory_Noetherian |
case intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
hn : Ideal.radical ⊥ ^ n ≤ ⊥
⊢ IsNilpotent (nilradical R) | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Ring.Idempotents
import Math... | exact ⟨n, eq_bot_iff.mpr hn⟩ | theorem IsNoetherianRing.isNilpotent_nilradical (R : Type*) [CommRing R] [IsNoetherianRing R] :
IsNilpotent (nilradical R) := by
obtain ⟨n, hn⟩ := Ideal.exists_radical_pow_le_of_fg (⊥ : Ideal R) (IsNoetherian.noetherian _)
| Mathlib.RingTheory.Noetherian.628_0.5UPGNrmhtW81IjE | theorem IsNoetherianRing.isNilpotent_nilradical (R : Type*) [CommRing R] [IsNoetherianRing R] :
IsNilpotent (nilradical R) | Mathlib_RingTheory_Noetherian |
C : Type u
inst✝¹ : Category.{v, u} C
α : Type u
inst✝ : SemilatticeSup α
X Y : α
f g : X ⟶ Y
⊢ f ≫ 𝟙 Y = g ≫ 𝟙 Y | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply ULift.ext | instance (priority := 100) isFilteredOrEmpty_of_semilatticeSup (α : Type u) [SemilatticeSup α] :
IsFilteredOrEmpty α
where
cocone_objs X Y := ⟨X ⊔ Y, homOfLE le_sup_left, homOfLE le_sup_right, trivial⟩
cocone_maps X Y f g := ⟨Y, 𝟙 _, by
| Mathlib.CategoryTheory.Filtered.Basic.89_0.dhnXC1TuYVuk8Vb | instance (priority | Mathlib_CategoryTheory_Filtered_Basic |
case h
C : Type u
inst✝¹ : Category.{v, u} C
α : Type u
inst✝ : SemilatticeSup α
X Y : α
f g : X ⟶ Y
⊢ (f ≫ 𝟙 Y).down = (g ≫ 𝟙 Y).down | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply Subsingleton.elim | instance (priority := 100) isFilteredOrEmpty_of_semilatticeSup (α : Type u) [SemilatticeSup α] :
IsFilteredOrEmpty α
where
cocone_objs X Y := ⟨X ⊔ Y, homOfLE le_sup_left, homOfLE le_sup_right, trivial⟩
cocone_maps X Y f g := ⟨Y, 𝟙 _, by
apply ULift.ext
| Mathlib.CategoryTheory.Filtered.Basic.89_0.dhnXC1TuYVuk8Vb | instance (priority | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝² : Category.{v, u} C
α : Type u
inst✝¹ : Preorder α
inst✝ : IsDirected α fun x x_1 => x ≤ x_1
X Y : α
f g : X ⟶ Y
⊢ f ≫ 𝟙 Y = g ≫ 𝟙 Y | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply ULift.ext | instance (priority := 100) isFilteredOrEmpty_of_directed_le (α : Type u) [Preorder α]
[IsDirected α (· ≤ ·)] : IsFilteredOrEmpty α where
cocone_objs X Y :=
let ⟨Z, h1, h2⟩ := exists_ge_ge X Y
⟨Z, homOfLE h1, homOfLE h2, trivial⟩
cocone_maps X Y f g := ⟨Y, 𝟙 _, by
| Mathlib.CategoryTheory.Filtered.Basic.102_0.dhnXC1TuYVuk8Vb | instance (priority | Mathlib_CategoryTheory_Filtered_Basic |
case h
C : Type u
inst✝² : Category.{v, u} C
α : Type u
inst✝¹ : Preorder α
inst✝ : IsDirected α fun x x_1 => x ≤ x_1
X Y : α
f g : X ⟶ Y
⊢ (f ≫ 𝟙 Y).down = (g ≫ 𝟙 Y).down | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply Subsingleton.elim | instance (priority := 100) isFilteredOrEmpty_of_directed_le (α : Type u) [Preorder α]
[IsDirected α (· ≤ ·)] : IsFilteredOrEmpty α where
cocone_objs X Y :=
let ⟨Z, h1, h2⟩ := exists_ge_ge X Y
⟨Z, homOfLE h1, homOfLE h2, trivial⟩
cocone_maps X Y f g := ⟨Y, 𝟙 _, by
apply ULift.ext
| Mathlib.CategoryTheory.Filtered.Basic.102_0.dhnXC1TuYVuk8Vb | instance (priority | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝² : Category.{v, u} C
α : Type u
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
⊢ IsFiltered α | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | infer_instance | example (α : Type u) [SemilatticeSup α] [OrderBot α] : IsFiltered α := by | Mathlib.CategoryTheory.Filtered.Basic.117_0.dhnXC1TuYVuk8Vb | example (α : Type u) [SemilatticeSup α] [OrderBot α] : IsFiltered α | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝² : Category.{v, u} C
α : Type u
inst✝¹ : SemilatticeSup α
inst✝ : OrderTop α
⊢ IsFiltered α | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | infer_instance | example (α : Type u) [SemilatticeSup α] [OrderTop α] : IsFiltered α := by | Mathlib.CategoryTheory.Filtered.Basic.119_0.dhnXC1TuYVuk8Vb | example (α : Type u) [SemilatticeSup α] [OrderTop α] : IsFiltered α | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝ : Category.{v, u} C
X Y : Discrete PUnit.{?u.7958 + 1}
⊢ X.as = { as := PUnit.unit }.as | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | trivial | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y := ⟨⟨PUnit.unit⟩, ⟨⟨by | Mathlib.CategoryTheory.Filtered.Basic.121_0.dhnXC1TuYVuk8Vb | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝ : Category.{v, u} C
X Y : Discrete PUnit.{?u.7958 + 1}
f g : X ⟶ Y
⊢ Y.as = { as := PUnit.unit }.as | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | trivial | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y := ⟨⟨PUnit.unit⟩, ⟨⟨by trivial⟩⟩, ⟨⟨Subsingleton.elim _ _⟩⟩, trivial⟩
cocone_maps X Y f g := ⟨⟨PUnit.unit⟩, ⟨⟨by | Mathlib.CategoryTheory.Filtered.Basic.121_0.dhnXC1TuYVuk8Vb | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝ : Category.{v, u} C
X Y : Discrete PUnit.{?u.7958 + 1}
f g : X ⟶ Y
⊢ f ≫ { down := { down := (_ : Y.as = Y.as) } } = g ≫ { down := { down := (_ : Y.as = Y.as) } } | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply ULift.ext | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y := ⟨⟨PUnit.unit⟩, ⟨⟨by trivial⟩⟩, ⟨⟨Subsingleton.elim _ _⟩⟩, trivial⟩
cocone_maps X Y f g := ⟨⟨PUnit.unit⟩, ⟨⟨by trivial⟩⟩, by
| Mathlib.CategoryTheory.Filtered.Basic.121_0.dhnXC1TuYVuk8Vb | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
case h
C : Type u
inst✝ : Category.{v, u} C
X Y : Discrete PUnit.{?u.7958 + 1}
f g : X ⟶ Y
⊢ (f ≫ { down := { down := (_ : Y.as = Y.as) } }).down = (g ≫ { down := { down := (_ : Y.as = Y.as) } }).down | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply Subsingleton.elim | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y := ⟨⟨PUnit.unit⟩, ⟨⟨by trivial⟩⟩, ⟨⟨Subsingleton.elim _ _⟩⟩, trivial⟩
cocone_maps X Y f g := ⟨⟨PUnit.unit⟩, ⟨⟨by trivial⟩⟩, by
apply ULift.ext
| Mathlib.CategoryTheory.Filtered.Basic.121_0.dhnXC1TuYVuk8Vb | instance : IsFiltered (Discrete PUnit) where
cocone_objs X Y | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : IsFilteredOrEmpty C
D : Type u₁
inst✝ : Category.{v₁, u₁} D
L : D ⥤ C
R : C ⥤ D
h : L ⊣ R
X Y : D
f g : X ⟶ Y
⊢ f ≫
(Adjunction.homEquiv h Y (coeq (?m.10744 h X Y f g) (?m.10745 h X Y f g)))
(coeqHom (?m.10744 h X Y f g) (?m.10745 h X Y f g)) =
g ≫
... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rw [← h.homEquiv_naturality_left, ← h.homEquiv_naturality_left, coeq_condition] | /-- If `C` is filtered or emtpy, and we have a functor `R : C ⥤ D` with a left adjoint, then `D` is
filtered or empty.
-/
theorem of_right_adjoint {L : D ⥤ C} {R : C ⥤ D} (h : L ⊣ R) : IsFilteredOrEmpty D :=
{ cocone_objs := fun X Y =>
⟨_, h.homEquiv _ _ (leftToMax _ _), h.homEquiv _ _ (rightToMax _ _), ⟨⟩⟩
... | Mathlib.CategoryTheory.Filtered.Basic.204_0.dhnXC1TuYVuk8Vb | /-- If `C` is filtered or emtpy, and we have a functor `R : C ⥤ D` with a left adjoint, then `D` is
filtered or empty.
-/
theorem of_right_adjoint {L : D ⥤ C} {R : C ⥤ D} (h : L ⊣ R) : IsFilteredOrEmpty D | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
⊢ ∃ S, ∀ {X : C}, X ∈ O → Nonempty (X ⟶ S) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· obtain ⟨S', w'⟩ := h
use max X S'
rintro Y mY
obtain rfl | h := eq_or_ne Y X
· exact ⟨leftToMax _ _⟩
· exact ⟨(w' (Finset.mem_of_mem_insert_of_ne mY h)).some ≫ rightToM... | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
| Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
⊢ ∃ S, ∀ {X : C}, X ∈ O → Nonempty (X ⟶ S) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | induction' O using Finset.induction with X O' nm h | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
| Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case empty
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
⊢ ∃ S, ∀ {X : C}, X ∈ ∅ → Nonempty (X ⟶ S) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩ | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
⊢ ∀ {X : C}, X ∈ ∅ → Nonempty (X ⟶ Classical.choice (_ : Nonempty C)) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | intro | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X✝ : C
⊢ X✝ ∈ ∅ → Nonempty (X✝ ⟶ Classical.choice (_ : Nonempty C)) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | simp | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case insert
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X : C
O' : Finset C
nm : X ∉ O'
h : ∃ S, ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S)
⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → Nonempty (X_1 ⟶ S) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain ⟨S', w'⟩ := h | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case insert.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X : C
O' : Finset C
nm : X ∉ O'
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → Nonempty (X_1 ⟶ S) | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | use max X S' | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· obtai... | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X : C
O' : Finset C
nm : X ∉ O'
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
⊢ ∀ {X_1 : C}, X_1 ∈ insert X O' → Nonempty (X_1 ⟶ max X S') | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rintro Y mY | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· obtai... | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X : C
O' : Finset C
nm : X ∉ O'
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
Y : C
mY : Y ∈ insert X O'
⊢ Nonempty (Y ⟶ max X S') | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain rfl | h := eq_or_ne Y X | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· obtai... | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case h.inl
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O' : Finset C
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
Y : C
nm : Y ∉ O'
mY : Y ∈ insert Y O'
⊢ Nonempty (Y ⟶ max Y S') | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact ⟨leftToMax _ _⟩ | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· obtai... | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
case h.inr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X : C
O' : Finset C
nm : X ∉ O'
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
Y : C
mY : Y ∈ insert X O'
h : Y ≠ X
⊢ Nonempty (Y ⟶ max X S') | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | exact ⟨(w' (Finset.mem_of_mem_insert_of_ne mY h)).some ≫ rightToMax _ _⟩ | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) := by
classical
induction' O using Finset.induction with X O' nm h
· exact ⟨Classical.choice IsFiltered.nonempty, by intro; simp⟩
· obtai... | Mathlib.CategoryTheory.Filtered.Basic.234_0.dhnXC1TuYVuk8Vb | /-- Any finite collection of objects in a filtered category has an object "to the right".
-/
theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
⊢ ∃ S T,
∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H → f ≫ T mY = ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | classical
induction' H using Finset.induction with h' H' nmf h''
· obtain ⟨S, f⟩ := sup_objs_exists O
refine' ⟨S, fun mX => (f mX).some, by rintro - - - - - ⟨⟩⟩
· obtain ⟨X, Y, mX, mY, f⟩ := h'
obtain ⟨S', T', w'⟩ := h''
refine' ⟨coeq (f ≫ T' mY) (T' mX), fun mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX),... | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
⊢ ∃ S T,
∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H → f ≫ T mY = ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | induction' H using Finset.induction with h' H' nmf h'' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case empty
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
⊢ ∃ S T,
∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain ⟨S, f⟩ := sup_objs_exists O | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case empty.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
S : C
f : ∀ {X : C}, X ∈ O → Nonempty (X ⟶ S)
⊢ ∃ S T,
∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
{ fst := X, snd := { fst := Y, snd := { fs... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | refine' ⟨S, fun mX => (f mX).some, by rintro - - - - - ⟨⟩⟩ | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
S : C
f : ∀ {X : C}, X ∈ O → Nonempty (X ⟶ S)
⊢ ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f_1 : X ⟶ Y},
{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY,... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rintro - - - - - ⟨⟩ | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case insert
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
h' : (X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)
H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
nmf : h' ∉ H'
h'' :
∃ ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain ⟨X, Y, mX, mY, f⟩ := h' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case insert.mk.mk.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
h'' :
∃ S T,
∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, s... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | obtain ⟨S', T', w'⟩ := h'' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case insert.mk.mk.mk.mk.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | refine' ⟨coeq (f ≫ T' mY) (T' mX), fun mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX), _⟩ | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case insert.mk.mk.mk.mk.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | intro X' Y' mX' mY' f' mf' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case insert.mk.mk.mk.mk.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rw [← Category.assoc] | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case insert.mk.mk.mk.mk.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | by_cases h : X = X' ∧ Y = Y' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → X ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | rcases h with ⟨rfl, rfl⟩ | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case pos.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | by_cases hf : f = f' | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → X ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | subst hf | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O : Finset C
H H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))
X Y : C
mX : X ∈ O
mY : Y ∈ O
f : X ⟶ Y
nmf : { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∉ H'
S' : C
T' : {X : C} → X ... | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryT... | apply coeq_condition | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib.CategoryTheory.Filtered.Basic.250_0.dhnXC1TuYVuk8Vb | /-- Given any `Finset` of objects `{X, ...}` and
indexed collection of `Finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem sup_exists :
∃ (S : C) (T : ∀ {X : C... | Mathlib_CategoryTheory_Filtered_Basic |
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