state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case refine'_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K J : Ideal S
hJ : J ∈ {J | K ≤ J ∧ IsMaximal J}
⊢ sInf {J | comap f K ≤ J ∧ IsMaximal J} ≤ comap f J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | haveI : J.IsMaximal := hJ.right | theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq... | Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN | theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson | Mathlib_RingTheory_JacobsonIdeal |
case refine'_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K J : Ideal S
hJ : J ∈ {J | K ≤ J ∧ IsMaximal J}
this : IsMaximal J
⊢ sInf {J | comap f K ≤ J ∧ IsMaximal J} ≤ comap f J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | refine' sInf_le ⟨comap_mono hJ.left, comap_isMaximal_of_surjective _ hf⟩ | theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq... | Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN | theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ I J : Ideal R
⊢ I ≤ J → jacobson I ≤ jacobson J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | intro h x hx | @[mono]
theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson := by
| Mathlib.RingTheory.JacobsonIdeal.230_0.Lz0MgLQMj1bGzuN | @[mono]
theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ I J : Ideal R
h : I ≤ J
x : R
hx : x ∈ jacobson I
⊢ x ∈ jacobson J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | erw [mem_sInf] at hx ⊢ | @[mono]
theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson := by
intro h x hx
| Mathlib.RingTheory.JacobsonIdeal.230_0.Lz0MgLQMj1bGzuN | @[mono]
theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ I J : Ideal R
h : I ≤ J
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ IsMaximal J} → x ∈ I_1
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J_1 | J ≤ J_1 ∧ IsMaximal J_1} → x ∈ I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact fun K ⟨hK, hK_max⟩ => hx ⟨Trans.trans h hK, hK_max⟩ | @[mono]
theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson := by
intro h x hx
erw [mem_sInf] at hx ⊢
| Mathlib.RingTheory.JacobsonIdeal.230_0.Lz0MgLQMj1bGzuN | @[mono]
theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
r : R
h : r - 1 ∈ jacobson ⊥
⊢ IsUnit r | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | cases' exists_mul_sub_mem_of_sub_one_mem_jacobson r h with s hs | theorem isUnit_of_sub_one_mem_jacobson_bot (r : R) (h : r - 1 ∈ jacobson (⊥ : Ideal R)) :
IsUnit r := by
| Mathlib.RingTheory.JacobsonIdeal.251_0.Lz0MgLQMj1bGzuN | theorem isUnit_of_sub_one_mem_jacobson_bot (r : R) (h : r - 1 ∈ jacobson (⊥ : Ideal R)) :
IsUnit r | Mathlib_RingTheory_JacobsonIdeal |
case intro
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
r : R
h : r - 1 ∈ jacobson ⊥
s : R
hs : s * r - 1 ∈ ⊥
⊢ IsUnit r | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [mem_bot, sub_eq_zero, mul_comm] at hs | theorem isUnit_of_sub_one_mem_jacobson_bot (r : R) (h : r - 1 ∈ jacobson (⊥ : Ideal R)) :
IsUnit r := by
cases' exists_mul_sub_mem_of_sub_one_mem_jacobson r h with s hs
| Mathlib.RingTheory.JacobsonIdeal.251_0.Lz0MgLQMj1bGzuN | theorem isUnit_of_sub_one_mem_jacobson_bot (r : R) (h : r - 1 ∈ jacobson (⊥ : Ideal R)) :
IsUnit r | Mathlib_RingTheory_JacobsonIdeal |
case intro
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
r : R
h : r - 1 ∈ jacobson ⊥
s : R
hs : r * s = 1
⊢ IsUnit r | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact isUnit_of_mul_eq_one _ _ hs | theorem isUnit_of_sub_one_mem_jacobson_bot (r : R) (h : r - 1 ∈ jacobson (⊥ : Ideal R)) :
IsUnit r := by
cases' exists_mul_sub_mem_of_sub_one_mem_jacobson r h with s hs
rw [mem_bot, sub_eq_zero, mul_comm] at hs
| Mathlib.RingTheory.JacobsonIdeal.251_0.Lz0MgLQMj1bGzuN | theorem isUnit_of_sub_one_mem_jacobson_bot (r : R) (h : r - 1 ∈ jacobson (⊥ : Ideal R)) :
IsUnit r | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
x : R
hx : x ∈ jacobson ⊥
y z : R
hz : z * y * x + z - 1 ∈ ⊥
⊢ (x * y + 1) * z = 1 | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rwa [add_mul, one_mul, ← sub_eq_zero, mul_right_comm, mul_comm _ z, mul_right_comm] | theorem mem_jacobson_bot {x : R} : x ∈ jacobson (⊥ : Ideal R) ↔ ∀ y, IsUnit (x * y + 1) :=
⟨fun hx y =>
let ⟨z, hz⟩ := (mem_jacobson_iff.1 hx) y
isUnit_iff_exists_inv.2
⟨z, by | Mathlib.RingTheory.JacobsonIdeal.258_0.Lz0MgLQMj1bGzuN | theorem mem_jacobson_bot {x : R} : x ∈ jacobson (⊥ : Ideal R) ↔ ∀ y, IsUnit (x * y + 1) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
x : R
h : ∀ (y : R), IsUnit (x * y + 1)
y b : R
hb : (x * y + 1) * b = 1
⊢ b * y * x + b - (x * y + 1) * b = 0 | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | ring | theorem mem_jacobson_bot {x : R} : x ∈ jacobson (⊥ : Ideal R) ↔ ∀ y, IsUnit (x * y + 1) :=
⟨fun hx y =>
let ⟨z, hz⟩ := (mem_jacobson_iff.1 hx) y
isUnit_iff_exists_inv.2
⟨z, by rwa [add_mul, one_mul, ← sub_eq_zero, mul_right_comm, mul_comm _ z, mul_right_comm]⟩,
fun h =>
mem_jacobson_iff.mpr fun ... | Mathlib.RingTheory.JacobsonIdeal.258_0.Lz0MgLQMj1bGzuN | theorem mem_jacobson_bot {x : R} : x ∈ jacobson (⊥ : Ideal R) ↔ ∀ y, IsUnit (x * y + 1) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
⊢ jacobson I = I ↔ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | have hf : Function.Surjective (Ideal.Quotient.mk I) := Submodule.Quotient.mk_surjective I | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
| Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
⊢ jacobson I = I ↔ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | constructor | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
⊢ jacobson I = I → jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | intro h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : jacobson I = I
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | replace h := congr_arg (Ideal.map (Ideal.Quotient.mk I)) h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : map (Quotient.mk I) (jacobson I) = map (Quotient.mk I) I
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [map_jacobson_of_surjective hf (le_of_eq mk_ker)] at h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : jacobson (map (Quotient.mk I) I) = map (Quotient.mk I) I
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simpa using h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
⊢ jacobson ⊥ = ⊥ → jacobson I = I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | intro h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : jacobson ⊥ = ⊥
⊢ jacobson I = I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | replace h := congr_arg (comap (Ideal.Quotient.mk I)) h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : comap (Quotient.mk I) (jacobson ⊥) = comap (Quotient.mk I) ⊥
⊢ jacobson I = I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [comap_jacobson_of_surjective hf, ← RingHom.ker_eq_comap_bot (Ideal.Quotient.mk I)] at h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : jacobson (RingHom.ker (Quotient.mk I)) = RingHom.ker (Quotient.mk I)
⊢ jacobson I = I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simpa using h | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ := by
have hf : Function.... | Mathlib.RingTheory.JacobsonIdeal.269_0.Lz0MgLQMj1bGzuN | /-- An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal -/
-- Porting note : changed `Quotient.mk'` to ``
theorem jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : Ideal (R ⧸ I)) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
⊢ radical I = jacobson I ↔ radical ⊥ = jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | have hf : Function.Surjective (Ideal.Quotient.mk I) := Submodule.Quotient.mk_surjective I | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
⊢ radical I = jacobson I ↔ radical ⊥ = jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | constructor | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
⊢ radical I = jacobson I → radical ⊥ = jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | intro h | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : radical I = jacobson I
⊢ radical ⊥ = jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | have := congr_arg (map (Ideal.Quotient.mk I)) h | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : radical I = jacobson I
this : map (Quotient.mk I) (radical I) = map (Quotient.mk I) (jacobson I)
⊢ radical ⊥ = jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [map_radical_of_surjective hf (le_of_eq mk_ker),
map_jacobson_of_surjective hf (le_of_eq mk_ker)] at this | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mp
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : radical I = jacobson I
this : radical (map (Quotient.mk I) I) = jacobson (map (Quotient.mk I) I)
⊢ radical ⊥ = jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simpa using this | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
⊢ radical ⊥ = jacobson ⊥ → radical I = jacobson I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | intro h | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : radical ⊥ = jacobson ⊥
⊢ radical I = jacobson I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | have := congr_arg (comap (Ideal.Quotient.mk I)) h | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : radical ⊥ = jacobson ⊥
this : comap (Quotient.mk I) (radical ⊥) = comap (Quotient.mk I) (jacobson ⊥)
⊢ radical I = jacobson I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [comap_radical, comap_jacobson_of_surjective hf,
← RingHom.ker_eq_comap_bot (Ideal.Quotient.mk I)] at this | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
case mpr
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : radical ⊥ = jacobson ⊥
this : radical (RingHom.ker (Quotient.mk I)) = jacobson (RingHom.ker (Quotient.mk I))
⊢ radical I = jacobson I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simpa using this | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib.RingTheory.JacobsonIdeal.286_0.Lz0MgLQMj1bGzuN | /-- The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide -/
-- Porting note : changed `Quotient.mk'` to ``
theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical ... | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
⊢ jacobson ⊥ ≤ sInf (map C '' {J | IsMaximal J}) | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | refine' le_sInf fun J => exists_imp.2 fun j hj => _ | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
| Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
⊢ jacobson ⊥ ≤ J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | haveI : j.IsMaximal := hj.1 | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
| Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
⊢ jacobson ⊥ ≤ J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J) | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
| Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
⊢ jacobson J = J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | suffices t : (⊥ : Ideal (Polynomial (R ⧸ j))).jacobson = ⊥ by
rw [← hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot]
replace t := congr_arg (map (polynomialQuotientEquivQuotientPolynomial j).toRingHom) t
rwa [map_jacobson_of_bijective _, map_bot] at t
exact RingEquiv.bijective (polynomialQuotientEquivQuo... | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
| Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
t : jacobson ⊥ = ⊥
⊢ jacobson J = J | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [← hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot] | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
t : jacobson ⊥ = ⊥
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | replace t := congr_arg (map (polynomialQuotientEquivQuotientPolynomial j).toRingHom) t | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
t :
map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) (jacobson ⊥) =
map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) ⊥
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rwa [map_jacobson_of_bijective _, map_bot] at t | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
t :
map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) (jacobson ⊥) =
map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) ⊥
⊢ Function.Bijective ... | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact RingEquiv.bijective (polynomialQuotientEquivQuotientPolynomial j) | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | refine' eq_bot_iff.2 fun f hf => _ | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
⊢ f ∈ ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | have r1 : (X : (R ⧸ j)[X]) ≠ 0 := fun hX => by
replace hX := congr_arg (fun f => coeff f 1) hX
simp only [coeff_X_one, coeff_zero] at hX
exact zero_ne_one hX.symm | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
hX : X = 0
⊢ False | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | replace hX := congr_arg (fun f => coeff f 1) hX | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
hX : (fun f => coeff f 1) X = (fun f => coeff f 1) 0
⊢ False | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simp only [coeff_X_one, coeff_zero] at hX | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
hX : 1 = 0
⊢ False | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact zero_ne_one hX.symm | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
r1 : X ≠ 0
⊢ f ∈ ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | have r2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit ((mem_jacobson_bot.1 hf) X)) | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
r1 : X ≠ 0
r2 : f * X + 1 = C (coeff (f * X + 1) 0)
⊢ f ∈ ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simp only [coeff_add, mul_coeff_zero, coeff_X_zero, mul_zero, coeff_one_zero, zero_add] at r2 | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
r1 : X ≠ 0
r2 : f * X + 1 = C 1
⊢ f ∈ ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | erw [add_left_eq_self] at r2 | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
J : Ideal R[X]
j : Ideal R
hj : j ∈ {J | IsMaximal J} ∧ map C j = J
this : IsMaximal j
f : (R ⧸ j)[X]
hf : f ∈ jacobson ⊥
r1 : X ≠ 0
r2 : f * X = 0
⊢ f ∈ ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | simpa using (mul_eq_zero.mp r2).resolve_right r1 | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by
refine' le_sInf fun J => exists_imp.2 fun j hj => _
haveI : j.IsMaximal := hj.1
refine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J)
suffic... | Mathlib.RingTheory.JacobsonIdeal.322_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
⊢ jacobson ⊥ = ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | refine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _) | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ := by
| Mathlib.RingTheory.JacobsonIdeal.346_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
⊢ sInf (map C '' {J | IsMaximal J}) ≤ ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | refine' fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ := by
refine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _)
| Mathlib.RingTheory.JacobsonIdeal.346_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
⊢ coeff f n = 0 | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | suffices f.coeff n ∈ Ideal.jacobson ⊥ by rwa [h, Submodule.mem_bot] at this | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ := by
refine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _)
refine' fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
Trans.trans (?_ : coeff f n = 0) (coe... | Mathlib.RingTheory.JacobsonIdeal.346_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
this : coeff f n ∈ jacobson ⊥
⊢ coeff f n = 0 | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rwa [h, Submodule.mem_bot] at this | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ := by
refine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _)
refine' fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
Trans.trans (?_ : coeff f n = 0) (coe... | Mathlib.RingTheory.JacobsonIdeal.346_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
⊢ coeff f n ∈ jacobson ⊥ | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact mem_sInf.2 fun j hj => (mem_map_C_iff.1 ((mem_sInf.1 hf) ⟨j, ⟨hj.2, rfl⟩⟩)) n | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ := by
refine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _)
refine' fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
Trans.trans (?_ : coeff f n = 0) (coe... | Mathlib.RingTheory.JacobsonIdeal.346_0.Lz0MgLQMj1bGzuN | theorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :
jacobson (⊥ : Ideal R[X]) = ⊥ | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
I : Ideal R
hi : IsLocal I
x : R
h : I ⊔ span {x} = ⊤
p : R
hpi : p ∈ I
q : R
hq : q ∈ span {x}
hpq : p + q = 1
r : R
hr : q = x * r
⊢ r * x - 1 ∈ I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [← hpq, mul_comm, ← hr, ← neg_sub, add_sub_cancel] | theorem IsLocal.mem_jacobson_or_exists_inv {I : Ideal R} (hi : IsLocal I) (x : R) :
x ∈ jacobson I ∨ ∃ y, y * x - 1 ∈ I :=
by_cases
(fun h : I ⊔ span {x} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h)
let ⟨r, hr⟩ := mem_span_singleton.1 hq
Or.inr ⟨r, by | Mathlib.RingTheory.JacobsonIdeal.384_0.Lz0MgLQMj1bGzuN | theorem IsLocal.mem_jacobson_or_exists_inv {I : Ideal R} (hi : IsLocal I) (x : R) :
x ∈ jacobson I ∨ ∃ y, y * x - 1 ∈ I | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
I : Ideal R
hi : IsLocal I
x : R
h : I ⊔ span {x} = ⊤
p : R
hpi : p ∈ I
q : R
hq : q ∈ span {x}
hpq : p + q = 1
r : R
hr : q = x * r
⊢ -p ∈ I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact I.neg_mem hpi | theorem IsLocal.mem_jacobson_or_exists_inv {I : Ideal R} (hi : IsLocal I) (x : R) :
x ∈ jacobson I ∨ ∃ y, y * x - 1 ∈ I :=
by_cases
(fun h : I ⊔ span {x} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h)
let ⟨r, hr⟩ := mem_span_singleton.1 hq
Or.inr ⟨r, by rw [←... | Mathlib.RingTheory.JacobsonIdeal.384_0.Lz0MgLQMj1bGzuN | theorem IsLocal.mem_jacobson_or_exists_inv {I : Ideal R} (hi : IsLocal I) (x : R) :
x ∈ jacobson I ∨ ∃ y, y * x - 1 ∈ I | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
I : Ideal R
hi : IsMaximal (radical I)
this : radical I = jacobson I
x y : R
hxy : x * y ∈ I
x✝ : ∃ y_1, y_1 * y - 1 ∈ I
z : R
hz : z * y - 1 ∈ I
⊢ x ∈ I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | rw [← mul_one x, ← sub_sub_cancel (z * y) 1, mul_sub, mul_left_comm] | theorem isPrimary_of_isMaximal_radical [CommRing R] {I : Ideal R} (hi : IsMaximal (radical I)) :
IsPrimary I :=
have : radical I = jacobson I :=
le_antisymm (le_sInf fun M ⟨him, hm⟩ => hm.isPrime.radical_le_iff.2 him)
(sInf_le ⟨le_radical, hi⟩)
⟨ne_top_of_lt <| lt_of_le_of_lt le_radical (lt_top_iff_ne... | Mathlib.RingTheory.JacobsonIdeal.398_0.Lz0MgLQMj1bGzuN | theorem isPrimary_of_isMaximal_radical [CommRing R] {I : Ideal R} (hi : IsMaximal (radical I)) :
IsPrimary I | Mathlib_RingTheory_JacobsonIdeal |
R : Type u
S : Type v
inst✝ : CommRing R
I : Ideal R
hi : IsMaximal (radical I)
this : radical I = jacobson I
x y : R
hxy : x * y ∈ I
x✝ : ∃ y_1, y_1 * y - 1 ∈ I
z : R
hz : z * y - 1 ∈ I
⊢ z * (x * y) - x * (z * y - 1) ∈ I | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib... | exact I.sub_mem (I.mul_mem_left _ hxy) (I.mul_mem_left _ hz) | theorem isPrimary_of_isMaximal_radical [CommRing R] {I : Ideal R} (hi : IsMaximal (radical I)) :
IsPrimary I :=
have : radical I = jacobson I :=
le_antisymm (le_sInf fun M ⟨him, hm⟩ => hm.isPrime.radical_le_iff.2 him)
(sInf_le ⟨le_radical, hi⟩)
⟨ne_top_of_lt <| lt_of_le_of_lt le_radical (lt_top_iff_ne... | Mathlib.RingTheory.JacobsonIdeal.398_0.Lz0MgLQMj1bGzuN | theorem isPrimary_of_isMaximal_radical [CommRing R] {I : Ideal R} (hi : IsMaximal (radical I)) :
IsPrimary I | Mathlib_RingTheory_JacobsonIdeal |
ι : Sort u_1
f : ι → ℕ
s : Set ℕ
⊢ ⨅ i, ↑(f i) ≠ ⊤ ↔ Nonempty ι | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.ENat.Basic
#align_import data.enat.lattice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062... | rw [Ne.def, iInf_coe_eq_top, not_isEmpty_iff] | lemma iInf_coe_ne_top : ⨅ i, (f i : ℕ∞) ≠ ⊤ ↔ Nonempty ι := by
| Mathlib.Data.ENat.Lattice.34_0.U3XTj6Gwuvfp15T | lemma iInf_coe_ne_top : ⨅ i, (f i : ℕ∞) ≠ ⊤ ↔ Nonempty ι | Mathlib_Data_ENat_Lattice |
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) → FG N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rintro ⟨t', h, rfl⟩ | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
| Mathlib.ModelTheory.FinitelyGenerated.45_0.mkqJR9tOk3JtWTX | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N | Mathlib_ModelTheory_FinitelyGenerated |
case intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
t' : Set M
h : Set.Finite t'
⊢ FG (LowerAdjoint.toFun (closure L) t') | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rcases Finite.exists_finset_coe h with ⟨t, rfl⟩ | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
| Mathlib.ModelTheory.FinitelyGenerated.45_0.mkqJR9tOk3JtWTX | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N | Mathlib_ModelTheory_FinitelyGenerated |
case intro.intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
t : Finset M
h : Set.Finite ↑t
⊢ FG (LowerAdjoint.toFun (closure L) ↑t) | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact ⟨t, rfl⟩ | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
| Mathlib.ModelTheory.FinitelyGenerated.45_0.mkqJR9tOk3JtWTX | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ FG N ↔ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [fg_def] | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
| Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) ↔ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | constructor | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
rw [fg_def]
| Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) → ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rintro ⟨S, Sfin, hS⟩ | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
rw [fg_def]
constructor
· | Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Sfin : Set.Finite S
hS : LowerAdjoint.toFun (closure L) S = N
⊢ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
| Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro.intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
n : ℕ
f : Fin n ↪ M
Sfin : Set.Finite (range ⇑f)
hS : LowerAdjoint.toFun (closure L) (range ⇑f) = N
⊢ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact ⟨n, f, hS⟩ | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
| Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (∃ n s, LowerAdjoint.toFun (closure L) (range s) = N) → ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rintro ⟨n, s, hs⟩ | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
exact ⟨n, f, hS⟩
· | Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr.intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
n : ℕ
s : Fin n → M
hs : LowerAdjoint.toFun (closure L) (range s) = N
⊢ ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | refine' ⟨range s, finite_range s, hs⟩ | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
exact ⟨n, f, hS⟩
· rintro ⟨n, s, hs⟩
| Mathlib.ModelTheory.FinitelyGenerated.52_0.mkqJR9tOk3JtWTX | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
⊢ LowerAdjoint.toFun (closure L) ↑∅ = ⊥ | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [Finset.coe_empty, closure_empty] | theorem fg_bot : (⊥ : L.Substructure M).FG :=
⟨∅, by | Mathlib.ModelTheory.FinitelyGenerated.63_0.mkqJR9tOk3JtWTX | theorem fg_bot : (⊥ : L.Substructure M).FG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
s : Set M
hs : Set.Finite s
⊢ LowerAdjoint.toFun (closure L) ↑(Finite.toFinset hs) = LowerAdjoint.toFun (closure L) s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [hs.coe_toFinset] | theorem fg_closure {s : Set M} (hs : s.Finite) : FG (closure L s) :=
⟨hs.toFinset, by | Mathlib.ModelTheory.FinitelyGenerated.67_0.mkqJR9tOk3JtWTX | theorem fg_closure {s : Set M} (hs : s.Finite) : FG (closure L s) | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N₁ N₂ : Substructure L M
hN₁ : FG N₁
hN₂ : FG N₂
t₁ : Set M
ht₁ : Set.Finite t₁ ∧ LowerAdjoint.toFun (closure L) t₁ = N₁
t₂ : Set M
ht₂ : Set.Finite t₂ ∧ LowerAdjoint.toFun (closure L) t₂ = N₂
⊢ LowerAdjoint.toFun (closure L) (t₁ ∪ t₂) = N₁ ⊔ N₂ | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [closure_union, ht₁.2, ht₂.2] | theorem FG.sup {N₁ N₂ : L.Substructure M} (hN₁ : N₁.FG) (hN₂ : N₂.FG) : (N₁ ⊔ N₂).FG :=
let ⟨t₁, ht₁⟩ := fg_def.1 hN₁
let ⟨t₂, ht₂⟩ := fg_def.1 hN₂
fg_def.2 ⟨t₁ ∪ t₂, ht₁.1.union ht₂.1, by | Mathlib.ModelTheory.FinitelyGenerated.75_0.mkqJR9tOk3JtWTX | theorem FG.sup {N₁ N₂ : L.Substructure M} (hN₁ : N₁.FG) (hN₂ : N₂.FG) : (N₁ ⊔ N₂).FG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M →[L] N
s : Substructure L M
hs : FG s
t : Set M
ht : Set.Finite t ∧ LowerAdjoint.toFun (closure L) t = s
⊢ LowerAdjoint.toFun (closure L) (⇑f '' t) = Substructure.map f s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [closure_image, ht.2] | theorem FG.map {N : Type*} [L.Structure N] (f : M →[L] N) {s : L.Substructure M} (hs : s.FG) :
(s.map f).FG :=
let ⟨t, ht⟩ := fg_def.1 hs
fg_def.2 ⟨f '' t, ht.1.image _, by | Mathlib.ModelTheory.FinitelyGenerated.81_0.mkqJR9tOk3JtWTX | theorem FG.map {N : Type*} [L.Structure N] (f : M →[L] N) {s : L.Substructure M} (hs : s.FG) :
(s.map f).FG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
hs : FG (Substructure.map (Embedding.toHom f) s)
⊢ FG s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rcases hs with ⟨t, h⟩ | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
| Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
⊢ FG s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [fg_def] | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
| Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
⊢ ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩ | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
| Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
⊢ LowerAdjoint.toFun (closure L) (⇑f ⁻¹' ↑t) = s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | have hf : Function.Injective f.toHom := f.injective | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
| Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
⊢ LowerAdjoint.toFun (closure L) (⇑f ⁻¹' ↑t) = s | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | refine' map_injective_of_injective hf _ | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
have hf : Function.Injective f.toHom := f.injective
| Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
⊢ Substructure.map (Embedding.toHom f) (LowerAdjoint.t... | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [← h, map_closure, Embedding.coe_toHom, image_preimage_eq_of_subset] | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injecti... | Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
⊢ ↑t ⊆ range ⇑f | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | intro x hx | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injecti... | Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
x : N
hx : x ∈ ↑t
⊢ x ∈ range ⇑f | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | have h' := subset_closure (L := L) hx | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injecti... | Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
x : N
hx : x ∈ ↑t
h' : x ∈ ↑(LowerAdjoint.toFun (closu... | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [h] at h' | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injecti... | Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
case intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
f : M ↪[L] N
s : Substructure L M
t : Finset N
h : LowerAdjoint.toFun (closure L) ↑t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
x : N
hx : x ∈ ↑t
h' : x ∈ ↑(Substructure.map (Embeddi... | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact Hom.map_le_range h' | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG := by
rcases hs with ⟨t, h⟩
rw [fg_def]
refine' ⟨f ⁻¹' t, t.finite_toSet.preimage (f.injective.injOn _), _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injecti... | Mathlib.ModelTheory.FinitelyGenerated.87_0.mkqJR9tOk3JtWTX | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
h : FG N
⊢ CG N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | obtain ⟨s, hf, rfl⟩ := fg_def.1 h | theorem FG.cg {N : L.Substructure M} (h : N.FG) : N.CG := by
| Mathlib.ModelTheory.FinitelyGenerated.111_0.mkqJR9tOk3JtWTX | theorem FG.cg {N : L.Substructure M} (h : N.FG) : N.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
s : Set M
hf : Set.Finite s
h : FG (LowerAdjoint.toFun (closure L) s)
⊢ CG (LowerAdjoint.toFun (closure L) s) | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | refine' ⟨s, hf.countable, rfl⟩ | theorem FG.cg {N : L.Substructure M} (h : N.FG) : N.CG := by
obtain ⟨s, hf, rfl⟩ := fg_def.1 h
| Mathlib.ModelTheory.FinitelyGenerated.111_0.mkqJR9tOk3JtWTX | theorem FG.cg {N : L.Substructure M} (h : N.FG) : N.CG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ CG N ↔ ↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [cg_def] | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
| Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = N) ↔
↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | constructor | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
| Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = N) →
↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rintro ⟨S, Scount, hS⟩ | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Scount : Set.Countable S
hS : LowerAdjoint.toFun (closure L) S = N
⊢ ↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rcases eq_empty_or_nonempty (N : Set M) with h | h | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
| Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro.inl
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Scount : Set.Countable S
hS : LowerAdjoint.toFun (closure L) S = N
h : ↑N = ∅
⊢ ↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact Or.intro_left _ h | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro.inr
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Scount : Set.Countable S
hS : LowerAdjoint.toFun (closure L) S = N
h : Set.Nonempty ↑N
⊢ ↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | obtain ⟨f, h'⟩ :=
(Scount.union (Set.countable_singleton h.some)).exists_eq_range
(singleton_nonempty h.some).inr | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
| Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro.inr.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Scount : Set.Countable S
hS : LowerAdjoint.toFun (closure L) S = N
h : Set.Nonempty ↑N
f : ℕ → M
h' : S ∪ {Set.Nonempty.some h} = range f
⊢ ↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | refine' Or.intro_right _ ⟨f, _⟩ | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro.inr.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Scount : Set.Countable S
hS : LowerAdjoint.toFun (closure L) S = N
h : Set.Nonempty ↑N
f : ℕ → M
h' : S ∪ {Set.Nonempty.some h} = range f
⊢ LowerAdjoint.toFun (closure L) (range f) = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [← h', closure_union, hS, sup_eq_left, closure_le] | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mp.intro.intro.inr.intro
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
S : Set M
Scount : Set.Countable S
hS : LowerAdjoint.toFun (closure L) S = N
h : Set.Nonempty ↑N
f : ℕ → M
h' : S ∪ {Set.Nonempty.some h} = range f
⊢ {Set.Nonempty.some h} ⊆ ↑N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact singleton_subset_iff.2 h.some_mem | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
⊢ (↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N) →
∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | intro h | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
h : ↑N = ∅ ∨ ∃ s, LowerAdjoint.toFun (closure L) (range s) = N
⊢ ∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | cases' h with h h | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr.inl
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
h : ↑N = ∅
⊢ ∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | refine' ⟨∅, countable_empty, closure_eq_of_le (empty_subset _) _⟩ | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr.inl
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
h : ↑N = ∅
⊢ N ≤ LowerAdjoint.toFun (closure L) ∅ | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [← SetLike.coe_subset_coe, h] | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr.inl
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
h : ↑N = ∅
⊢ ∅ ⊆ ↑(LowerAdjoint.toFun (closure L) ∅) | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact empty_subset _ | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr.inr
L : Language
M : Type u_1
inst✝ : Structure L M
N : Substructure L M
h : ∃ s, LowerAdjoint.toFun (closure L) (range s) = N
⊢ ∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = N | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | obtain ⟨f, rfl⟩ := h | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
case mpr.inr.intro
L : Language
M : Type u_1
inst✝ : Structure L M
f : ℕ → M
⊢ ∃ S, Set.Countable S ∧ LowerAdjoint.toFun (closure L) S = LowerAdjoint.toFun (closure L) (range f) | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | exact ⟨range f, countable_range _, rfl⟩ | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(S... | Mathlib.ModelTheory.FinitelyGenerated.116_0.mkqJR9tOk3JtWTX | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
N₁ N₂ : Substructure L M
hN₁ : CG N₁
hN₂ : CG N₂
t₁ : Set M
ht₁ : Set.Countable t₁ ∧ LowerAdjoint.toFun (closure L) t₁ = N₁
t₂ : Set M
ht₂ : Set.Countable t₂ ∧ LowerAdjoint.toFun (closure L) t₂ = N₂
⊢ LowerAdjoint.toFun (closure L) (t₁ ∪ t₂) = N₁ ⊔ N₂ | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"... | rw [closure_union, ht₁.2, ht₂.2] | theorem CG.sup {N₁ N₂ : L.Substructure M} (hN₁ : N₁.CG) (hN₂ : N₂.CG) : (N₁ ⊔ N₂).CG :=
let ⟨t₁, ht₁⟩ := cg_def.1 hN₁
let ⟨t₂, ht₂⟩ := cg_def.1 hN₂
cg_def.2 ⟨t₁ ∪ t₂, ht₁.1.union ht₂.1, by | Mathlib.ModelTheory.FinitelyGenerated.150_0.mkqJR9tOk3JtWTX | theorem CG.sup {N₁ N₂ : L.Substructure M} (hN₁ : N₁.CG) (hN₂ : N₂.CG) : (N₁ ⊔ N₂).CG | Mathlib_ModelTheory_FinitelyGenerated |
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