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 |
|---|---|---|---|---|---|---|
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 : CG s
t : Set M
ht : Set.Countable 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 CG.map {N : Type*} [L.Structure N] (f : M →[L] N) {s : L.Substructure M} (hs : s.CG) :
(s.map f).CG :=
let ⟨t, ht⟩ := cg_def.1 hs
cg_def.2 ⟨f '' t, ht.1.image _, by | Mathlib.ModelTheory.FinitelyGenerated.156_0.mkqJR9tOk3JtWTX | theorem CG.map {N : Type*} [L.Structure N] (f : M →[L] N) {s : L.Substructure M} (hs : s.CG) :
(s.map f).CG | 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 : CG (Substructure.map (Embedding.toHom f) s)
⊢ CG 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, h1, h2⟩ | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
| Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
⊢ CG 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 [cg_def] | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
| Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
⊢ ∃ S, Set.Countable 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, h1.preimage f.injective, _⟩ | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
| Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : 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 CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
| Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
⊢ LowerAdjoint.toFun (closure ... | /-
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 CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
have hf : Function.Injective f.toHom := f.injective
| Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
⊢ Substructure.map (Embedding.... | /-
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 [← h2, map_closure, Embedding.coe_toHom, image_preimage_eq_of_subset] | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injective_of_injective h... | Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : 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 CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injective_of_injective h... | Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
x : N
hx : x ∈ t
⊢ x ∈ range ⇑... | /-
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 CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injective_of_injective h... | Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
x : N
hx : x ∈ t
h' : x ∈ ↑(Lo... | /-
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 [h2] at h' | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injective_of_injective h... | Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
case intro.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 : Set N
h1 : Set.Countable t
h2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s
hf : Function.Injective ⇑(Embedding.toHom f)
x : N
hx : x ∈ t
h' : x ∈ ↑(Su... | /-
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 CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG := by
rcases hs with ⟨t, h1, h2⟩
rw [cg_def]
refine' ⟨f ⁻¹' t, h1.preimage f.injective, _⟩
have hf : Function.Injective f.toHom := f.injective
refine' map_injective_of_injective h... | Mathlib.ModelTheory.FinitelyGenerated.162_0.mkqJR9tOk3JtWTX | theorem CG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).CG) : s.CG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
inst✝ : Countable ((l : ℕ) × Functions L l)
s : Substructure L M
⊢ CG s ↔ Countable ↥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' ⟨_, fun h => ⟨s, h.to_set, s.closure_eq⟩⟩ | theorem cg_iff_countable [Countable (Σl, L.Functions l)] {s : L.Substructure M} :
s.CG ↔ Countable s := by
| Mathlib.ModelTheory.FinitelyGenerated.176_0.mkqJR9tOk3JtWTX | theorem cg_iff_countable [Countable (Σl, L.Functions l)] {s : L.Substructure M} :
s.CG ↔ Countable s | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
inst✝ : Countable ((l : ℕ) × Functions L l)
s : Substructure L M
⊢ CG s → Countable ↥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"... | rintro ⟨s, h, rfl⟩ | theorem cg_iff_countable [Countable (Σl, L.Functions l)] {s : L.Substructure M} :
s.CG ↔ Countable s := by
refine' ⟨_, fun h => ⟨s, h.to_set, s.closure_eq⟩⟩
| Mathlib.ModelTheory.FinitelyGenerated.176_0.mkqJR9tOk3JtWTX | theorem cg_iff_countable [Countable (Σl, L.Functions l)] {s : L.Substructure M} :
s.CG ↔ Countable s | Mathlib_ModelTheory_FinitelyGenerated |
case intro.intro
L : Language
M : Type u_1
inst✝¹ : Structure L M
inst✝ : Countable ((l : ℕ) × Functions L l)
s : Set M
h : Set.Countable s
⊢ Countable ↥(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"... | exact h.substructure_closure L | theorem cg_iff_countable [Countable (Σl, L.Functions l)] {s : L.Substructure M} :
s.CG ↔ Countable s := by
refine' ⟨_, fun h => ⟨s, h.to_set, s.closure_eq⟩⟩
rintro ⟨s, h, rfl⟩
| Mathlib.ModelTheory.FinitelyGenerated.176_0.mkqJR9tOk3JtWTX | theorem cg_iff_countable [Countable (Σl, L.Functions l)] {s : L.Substructure M} :
s.CG ↔ Countable s | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
⊢ FG L M ↔ ∃ S, Set.Finite S ∧ 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 [fg_def, Substructure.fg_def] | /-- An equivalent expression of `Structure.FG` in terms of `Set.Finite` instead of `Finset`. -/
theorem fg_iff : FG L M ↔ ∃ S : Set M, S.Finite ∧ closure L S = (⊤ : L.Substructure M) := by
| Mathlib.ModelTheory.FinitelyGenerated.209_0.mkqJR9tOk3JtWTX | /-- An equivalent expression of `Structure.FG` in terms of `Set.Finite` instead of `Finset`. -/
theorem fg_iff : FG L M ↔ ∃ S : Set M, S.Finite ∧ closure L S = (⊤ : L.Substructure M) | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : FG L M
f : M →[L] N
⊢ Substructure.FG (Hom.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"... | rw [Hom.range_eq_map] | theorem FG.range {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N) : f.range.FG := by
| Mathlib.ModelTheory.FinitelyGenerated.214_0.mkqJR9tOk3JtWTX | theorem FG.range {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N) : f.range.FG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : FG L M
f : M →[L] N
⊢ Substructure.FG (map 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 (fg_def.1 h).map f | theorem FG.range {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N) : f.range.FG := by
rw [Hom.range_eq_map]
| Mathlib.ModelTheory.FinitelyGenerated.214_0.mkqJR9tOk3JtWTX | theorem FG.range {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N) : f.range.FG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : FG L M
f : M →[L] N
hs : Function.Surjective ⇑f
⊢ FG L 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 [← Hom.range_eq_top] at hs | theorem FG.map_of_surjective {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N)
(hs : Function.Surjective f) : FG L N := by
| Mathlib.ModelTheory.FinitelyGenerated.219_0.mkqJR9tOk3JtWTX | theorem FG.map_of_surjective {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N)
(hs : Function.Surjective f) : FG L N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : FG L M
f : M →[L] N
hs : Hom.range f = ⊤
⊢ FG L 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, ← hs] | theorem FG.map_of_surjective {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N)
(hs : Function.Surjective f) : FG L N := by
rw [← Hom.range_eq_top] at hs
| Mathlib.ModelTheory.FinitelyGenerated.219_0.mkqJR9tOk3JtWTX | theorem FG.map_of_surjective {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N)
(hs : Function.Surjective f) : FG L N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : FG L M
f : M →[L] N
hs : Hom.range f = ⊤
⊢ Substructure.FG (Hom.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 h.range f | theorem FG.map_of_surjective {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N)
(hs : Function.Surjective f) : FG L N := by
rw [← Hom.range_eq_top] at hs
rw [fg_def, ← hs]
| Mathlib.ModelTheory.FinitelyGenerated.219_0.mkqJR9tOk3JtWTX | theorem FG.map_of_surjective {N : Type*} [L.Structure N] (h : FG L M) (f : M →[L] N)
(hs : Function.Surjective f) : FG L N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
⊢ CG L M ↔ ∃ S, Set.Countable S ∧ 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 [cg_def, Substructure.cg_def] | /-- An equivalent expression of `Structure.cg`. -/
theorem cg_iff : CG L M ↔ ∃ S : Set M, S.Countable ∧ closure L S = (⊤ : L.Substructure M) := by
| Mathlib.ModelTheory.FinitelyGenerated.230_0.mkqJR9tOk3JtWTX | /-- An equivalent expression of `Structure.cg`. -/
theorem cg_iff : CG L M ↔ ∃ S : Set M, S.Countable ∧ closure L S = (⊤ : L.Substructure M) | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : CG L M
f : M →[L] N
⊢ Substructure.CG (Hom.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"... | rw [Hom.range_eq_map] | theorem CG.range {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N) : f.range.CG := by
| Mathlib.ModelTheory.FinitelyGenerated.235_0.mkqJR9tOk3JtWTX | theorem CG.range {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N) : f.range.CG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : CG L M
f : M →[L] N
⊢ Substructure.CG (map 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 (cg_def.1 h).map f | theorem CG.range {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N) : f.range.CG := by
rw [Hom.range_eq_map]
| Mathlib.ModelTheory.FinitelyGenerated.235_0.mkqJR9tOk3JtWTX | theorem CG.range {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N) : f.range.CG | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : CG L M
f : M →[L] N
hs : Function.Surjective ⇑f
⊢ CG L 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 [← Hom.range_eq_top] at hs | theorem CG.map_of_surjective {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N)
(hs : Function.Surjective f) : CG L N := by
| Mathlib.ModelTheory.FinitelyGenerated.240_0.mkqJR9tOk3JtWTX | theorem CG.map_of_surjective {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N)
(hs : Function.Surjective f) : CG L N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : CG L M
f : M →[L] N
hs : Hom.range f = ⊤
⊢ CG L 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, ← hs] | theorem CG.map_of_surjective {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N)
(hs : Function.Surjective f) : CG L N := by
rw [← Hom.range_eq_top] at hs
| Mathlib.ModelTheory.FinitelyGenerated.240_0.mkqJR9tOk3JtWTX | theorem CG.map_of_surjective {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N)
(hs : Function.Surjective f) : CG L N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
N : Type u_2
inst✝ : Structure L N
h : CG L M
f : M →[L] N
hs : Hom.range f = ⊤
⊢ Substructure.CG (Hom.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 h.range f | theorem CG.map_of_surjective {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N)
(hs : Function.Surjective f) : CG L N := by
rw [← Hom.range_eq_top] at hs
rw [cg_def, ← hs]
| Mathlib.ModelTheory.FinitelyGenerated.240_0.mkqJR9tOk3JtWTX | theorem CG.map_of_surjective {N : Type*} [L.Structure N] (h : CG L M) (f : M →[L] N)
(hs : Function.Surjective f) : CG L N | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝¹ : Structure L M
inst✝ : Countable ((l : ℕ) × Functions L l)
⊢ CG L M ↔ Countable M | /-
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, Substructure.cg_iff_countable, topEquiv.toEquiv.countable_iff] | theorem cg_iff_countable [Countable (Σl, L.Functions l)] : CG L M ↔ Countable M := by
| Mathlib.ModelTheory.FinitelyGenerated.247_0.mkqJR9tOk3JtWTX | theorem cg_iff_countable [Countable (Σl, L.Functions l)] : CG L M ↔ Countable M | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
⊢ FG S ↔ Structure.FG 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 [Structure.fg_def] | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
| Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
⊢ FG S ↔ FG ⊤ | /-
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' ⟨fun h => FG.of_map_embedding S.subtype _, fun h => _⟩ | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
rw [Structure.fg_def]
| Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_1
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h : FG S
⊢ FG (map (Embedding.toHom (subtype 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 [← Hom.range_eq_map, range_subtype] | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
rw [Structure.fg_def]
refine' ⟨fun h => FG.of_map_embedding S.subtype _, fun h => _⟩
· | Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_1
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h : FG 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"... | exact h | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
rw [Structure.fg_def]
refine' ⟨fun h => FG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
| Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_2
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h : FG ⊤
⊢ 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"... | have h := h.map S.subtype.toHom | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
rw [Structure.fg_def]
refine' ⟨fun h => FG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· | Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_2
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h✝ : FG ⊤
h : FG (map (Embedding.toHom (subtype 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 [← Hom.range_eq_map, range_subtype] at h | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
rw [Structure.fg_def]
refine' ⟨fun h => FG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· have h := h.map S.subtype.toHom
| Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_2
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h✝ : FG ⊤
h : FG 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"... | exact h | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S := by
rw [Structure.fg_def]
refine' ⟨fun h => FG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· have h := h.map S.subtype.toHom
rw [← Hom.range_eq_map, range_subtype] a... | Mathlib.ModelTheory.FinitelyGenerated.267_0.mkqJR9tOk3JtWTX | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
⊢ CG S ↔ Structure.CG 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 [Structure.cg_def] | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
| Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
⊢ CG S ↔ CG ⊤ | /-
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' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩ | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
rw [Structure.cg_def]
| Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_1
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h : CG S
⊢ CG (map (Embedding.toHom (subtype 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 [← Hom.range_eq_map, range_subtype] | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
rw [Structure.cg_def]
refine' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩
· | Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_1
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h : CG S
⊢ CG 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"... | exact h | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
rw [Structure.cg_def]
refine' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
| Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_2
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h : CG ⊤
⊢ CG 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 h := h.map S.subtype.toHom | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
rw [Structure.cg_def]
refine' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· | Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_2
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h✝ : CG ⊤
h : CG (map (Embedding.toHom (subtype S)) ⊤)
⊢ CG 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 [← Hom.range_eq_map, range_subtype] at h | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
rw [Structure.cg_def]
refine' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· have h := h.map S.subtype.toHom
| Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
case refine'_2
L : Language
M : Type u_1
inst✝ : Structure L M
S : Substructure L M
h✝ : CG ⊤
h : CG S
⊢ CG 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"... | exact h | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S := by
rw [Structure.cg_def]
refine' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩
· rw [← Hom.range_eq_map, range_subtype]
exact h
· have h := h.map S.subtype.toHom
rw [← Hom.range_eq_map, range_subtype] a... | Mathlib.ModelTheory.FinitelyGenerated.284_0.mkqJR9tOk3JtWTX | theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S | Mathlib_ModelTheory_FinitelyGenerated |
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
⊢ ∃ α, F⟮α⟯ = ⊤ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
| Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
⊢ ∃ α, F⟮α⟯ = ⊤ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | use α | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
| Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case h
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
⊢ F⟮↑α⟯ = ⊤ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | apply eq_top_iff.mpr | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
| Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case h
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
⊢ ⊤ ≤ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rintro x - | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
| Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case h
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
⊢ x ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | by_cases hx : x = 0 | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
| Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case pos
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : x = 0
⊢ x ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [hx] | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case pos
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : x = 0
⊢ 0 ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact F⟮α.val⟯.zero_mem | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
| Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case neg
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : ¬x = 0
⊢ x ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx)) | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case neg.intro
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : ¬x = 0
n : ℤ
hn : (fun x => α ^ x) n = Units.mk0 x hx
⊢ x ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | simp only at hn | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn... | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case neg.intro
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : ¬x = 0
n : ℤ
hn : α ^ n = Units.mk0 x hx
⊢ x ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]] | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn... | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : ¬x = 0
n : ℤ
hn : α ^ n = Units.mk0 x hx
⊢ x = ↑α ^ n | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | norm_cast | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn... | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : ¬x = 0
n : ℤ
hn : α ^ n = Units.mk0 x hx
⊢ x = ↑(α ^ n) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [hn, Units.val_mk0] | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn... | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
case neg.intro
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite E
α : Eˣ
hα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α
x : E
hx : ¬x = 0
n : ℤ
hn : α ^ n = Units.mk0 x hx
⊢ ↑α ^ n ∈ F⟮↑α⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn... | Mathlib.FieldTheory.PrimitiveElement.56_0.R5HND7n71i1v1rZ | /-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
⊢ ∃ c, ∀ α' ∈ roots (Polynomial.map ϕ f), ∀ β' ∈ roots (Polynomial.map ϕ g), -(α' - α) / (β' - β) ≠ ϕ c | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let sf := (f.map ϕ).roots | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
| Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
⊢ ∃ c, ∀ α' ∈ roots (Polynomial.map ϕ f), ∀ β' ∈ roots (Polynomial.map ϕ g), -(α' - α) / (β' - β) ≠ ϕ c | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let sg := (g.map ϕ).roots | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
| Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
sg : Multiset E := roots (Polynomial.map ϕ g)
⊢ ∃ c, ∀ α' ∈ roots (Polynomial.map ϕ f), ∀ β' ∈ roots (Polynomial.map ϕ g), -(α' - α) / (β' - β) ≠ ϕ c | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
| Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
sg : Multiset E := roots (Polynomial.map ϕ g)
s : Finset E := Multiset.toFinset (Multiset.bind sf fun α' => Multiset.map (fun β' => -(α' - α) / (β' - β)) sg)
⊢ ∃ c,... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
| Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
sg : Multiset E := roots (Polynomial.map ϕ g)
s : Finset E := Multiset.toFinset (Multiset.bind sf fun α' => Multiset.map (fun β' => -(α' - α) / (β' - β)) sg)
s' : F... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s' | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y... | Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
sg : Multiset E := roots (Polynomial.map ϕ g)
s : Finset E := Multiset.toFinset (Multiset.bind sf fun α' => Multiset.map (fun β' => -(α' - α) / (β' - β))... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y... | Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
sg : Multiset E := roots (Polynomial.map ϕ g)
s : Finset E := Multiset.toFinset (Multiset.bind sf fun α' => Multiset.map (fun β' => -(α' - α) / (β' - β))... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | push_neg at hc | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y... | Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝² : Field F
inst✝¹ : Infinite F
E : Type u_2
inst✝ : Field E
ϕ : F →+* E
α β : E
f g : F[X]
sf : Multiset E := roots (Polynomial.map ϕ f)
sg : Multiset E := roots (Polynomial.map ϕ g)
s : Finset E := Multiset.toFinset (Multiset.bind sf fun α' => Multiset.map (fun β' => -(α' - α) / (β' - β))... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact ⟨c, hc⟩ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y... | Mathlib.FieldTheory.PrimitiveElement.87_0.R5HND7n71i1v1rZ | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have hα := IsSeparable.isIntegral F α | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
| Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have hβ := IsSeparable.isIntegral F β | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
| Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let f := minpoly F α | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let g := minpoly F β | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let ιFE := algebraMap F E | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let ιEE' := algebraMap E (SplittingField (g.map ιFE)) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := a... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let γ := α + c • β | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | suffices β_in_Fγ : β ∈ F⟮γ⟯ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | use γ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE ... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | apply le_antisymm | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [adjoin_le_iff] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := a... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [← add_sub_cancel α (c • β)] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := a... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rintro x (rfl | rfl) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a.inl
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hβ : IsIntegral F β
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Poly... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | assumption | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a.inr
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
f : F[X] := minpoly F α
ιFE : F →+* E := algebraMap F E
c : F
x : E
hβ : IsIntegral F x
g : F[X] := minpoly F x
ιEE' : E →+* SplittingField (Pol... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | assumption | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [adjoin_simple_le_iff] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β}) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιF... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯)) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
... | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := a... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := a... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [finale] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.ma... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact Subtype.mem (-p.coeff 0 / p.coeff 1) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case p_linear
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.m... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case p_linear
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.m... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := a... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | apply eval_gcd_eq_zero | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case hf
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case hg
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [eval_map, ← aeval_def, minpoly.aeval] | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
case p_linear
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.m... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _) | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
h... | Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ | /-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | Mathlib_FieldTheory_PrimitiveElement |
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