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 |
|---|---|---|---|---|---|---|
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
x : P
hx : x = 0
⊢ (Algebra.linearMap R P) 0 = x | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp [hx] | @[simp]
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 :=
⟨fun ⟨x', x'_mem_zero, x'_eq_x⟩ => by
have x'_eq_zero : x' = 0 := x'_mem_zero
simp [x'_eq_x.symm, x'_eq_zero], fun hx => ⟨0, rfl, by | Mathlib.RingTheory.FractionalIdeal.275_0.90B1BH8AtSmfl9S | @[simp]
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
h : ↑I = ⊥
⊢ (fun I => ↑I) I = (fun I => ↑I) 0 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp [h] | theorem coeToSubmodule_eq_bot {I : FractionalIdeal S P} : (I : Submodule R P) = ⊥ ↔ I = 0 :=
⟨fun h => coeToSubmodule_injective (by | Mathlib.RingTheory.FractionalIdeal.326_0.90B1BH8AtSmfl9S | theorem coeToSubmodule_eq_bot {I : FractionalIdeal S P} : (I : Submodule R P) = ⊥ ↔ I = 0 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
h : I = 0
⊢ ↑I = ⊥ | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp [h] | theorem coeToSubmodule_eq_bot {I : FractionalIdeal S P} : (I : Submodule R P) = ⊥ ↔ I = 0 :=
⟨fun h => coeToSubmodule_injective (by simp [h]), fun h => by | Mathlib.RingTheory.FractionalIdeal.326_0.90B1BH8AtSmfl9S | theorem coeToSubmodule_eq_bot {I : FractionalIdeal S P} : (I : Submodule R P) = ⊥ ↔ I = 0 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
⊢ ↑1 = 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [coe_one_eq_coeSubmodule_top, coeSubmodule_top] | @[simp, norm_cast]
theorem coe_one : (↑(1 : FractionalIdeal S P) : Submodule R P) = 1 := by
| Mathlib.RingTheory.FractionalIdeal.369_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem coe_one : (↑(1 : FractionalIdeal S P) : Submodule R P) = 1 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
⊢ 0 ≤ I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro x hx | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I := by
| Mathlib.RingTheory.FractionalIdeal.390_0.90B1BH8AtSmfl9S | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
x : P
hx : x ∈ 0
⊢ x ∈ I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [(mem_zero_iff _).mp hx] | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I := by
intro x hx
-- Porting note: changed the proof from convert; simp into rw; exact
| Mathlib.RingTheory.FractionalIdeal.390_0.90B1BH8AtSmfl9S | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
x : P
hx : x ∈ 0
⊢ 0 ∈ I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact zero_mem (I : Submodule R P) | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I := by
intro x hx
-- Porting note: changed the proof from convert; simp into rw; exact
rw [(mem_zero_iff _).mp hx]
| Mathlib.RingTheory.FractionalIdeal.390_0.90B1BH8AtSmfl9S | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
h : I = 0
x : P
hx : x ∈ I
⊢ x = 0 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simpa [h, mem_zero_iff] using hx | theorem eq_zero_iff {I : FractionalIdeal S P} : I = 0 ↔ ∀ x ∈ I, x = (0 : P) :=
⟨fun h x hx => by | Mathlib.RingTheory.FractionalIdeal.412_0.90B1BH8AtSmfl9S | theorem eq_zero_iff {I : FractionalIdeal S P} : I = 0 ↔ ∀ x ∈ I, x = (0 : P) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩ | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
| Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : b... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [smul_add] | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
| Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : b... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | apply isInteger_add | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
| Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ ... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [mul_smul, smul_comm] | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· | Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ ... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact isInteger_smul (hI bI hbI) | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.intro.hb
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ ... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [mul_smul] | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.intro.hb
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ ... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact isInteger_smul (hJ bJ hbJ) | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib.RingTheory.FractionalIdeal.417_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [m... | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
J : Submodule R P
b : P
hb : b ∈ I ⊓ J
⊢ IsInteger R (aI • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rcases mem_inf.mp hb with ⟨hbI, _⟩ | theorem _root_.IsFractional.inf_right {I : Submodule R P} :
IsFractional S I → ∀ J, IsFractional S (I ⊓ J)
| ⟨aI, haI, hI⟩, J =>
⟨aI, haI, fun b hb => by
| Mathlib.RingTheory.FractionalIdeal.430_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.inf_right {I : Submodule R P} :
IsFractional S I → ∀ J, IsFractional S (I ⊓ J)
| ⟨aI, haI, hI⟩, J =>
⟨aI, haI, fun b hb => by
rcases mem_inf.mp hb with ⟨hbI, _⟩
exact hI b hbI⟩ | Mathlib_RingTheory_FractionalIdeal |
case intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
J : Submodule R P
b : P
hb : b ∈ I ⊓ J
hbI : b ∈ I
right✝ : b ∈ J
⊢ IsInteger R (aI • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact hI b hbI | theorem _root_.IsFractional.inf_right {I : Submodule R P} :
IsFractional S I → ∀ J, IsFractional S (I ⊓ J)
| ⟨aI, haI, hI⟩, J =>
⟨aI, haI, fun b hb => by
rcases mem_inf.mp hb with ⟨hbI, _⟩
| Mathlib.RingTheory.FractionalIdeal.430_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.inf_right {I : Submodule R P} :
IsFractional S I → ∀ J, IsFractional S (I ⊓ J)
| ⟨aI, haI, hI⟩, J =>
⟨aI, haI, fun b hb => by
rcases mem_inf.mp hb with ⟨hbI, _⟩
exact hI b hbI⟩ | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ IsFractional S (0 • I) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [zero_smul] | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
| Mathlib.RingTheory.FractionalIdeal.483_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
rw [succ_nsmul]
exact h.sup (IsFractional.nsmul n h) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ IsFractional S 0 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | convert ((0 : Ideal R) : FractionalIdeal S P).isFractional | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
| Mathlib.RingTheory.FractionalIdeal.483_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
rw [succ_nsmul]
exact h.sup (IsFractional.nsmul n h) | Mathlib_RingTheory_FractionalIdeal |
case h.e'_7
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ 0 = ↑↑0 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
| Mathlib.RingTheory.FractionalIdeal.483_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
rw [succ_nsmul]
exact h.sup (IsFractional.nsmul n h) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
n : ℕ
h : IsFractional S I
⊢ IsFractional S ((n + 1) • I) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [succ_nsmul] | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
| Mathlib.RingTheory.FractionalIdeal.483_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
rw [succ_nsmul]
exact h.sup (IsFractional.nsmul n h) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
n : ℕ
h : IsFractional S I
⊢ IsFractional S (I + n • I) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact h.sup (IsFractional.nsmul n h) | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
rw [succ_nsmul]
| Mathlib.RingTheory.FractionalIdeal.483_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.nsmul {I : Submodule R P} :
∀ n : ℕ, IsFractional S I → IsFractional S (n • I : Submodule R P)
| 0, _ => by
rw [zero_smul]
convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
simp
| n + 1, h => by
rw [succ_nsmul]
exact h.sup (IsFractional.nsmul n h) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
⊢ IsInteger R ((aI * aJ) • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | refine Submodule.mul_induction_on hb ?_ ?_ | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
| Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
⊢ ∀ m ∈ I, ∀ n ∈ J, IsIntege... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro m hm n hn | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P
hn : ... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨n', hn'⟩ := hJ n hn | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
| Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_1.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← Algebra.smul_def] | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ :... | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_1.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | apply hI | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ :... | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_1.intro.a
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact Submodule.smul_mem _ _ hm | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ :... | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_2
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
⊢ ∀ (x y : P), IsInteger R (... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro x y hx hy | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ :... | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_2
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
x y : P
hx : IsInteger R ((a... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [smul_add] | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ :... | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
case refine_2
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
x y : P
hx : IsInteger R ((a... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | apply isInteger_add hx hy | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ :... | Mathlib.RingTheory.FractionalIdeal.502_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : FractionalIdeal S P
⊢ I * J = { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [← mul_eq_mul, mul] | theorem mul_def (I J : FractionalIdeal S P) : I * J = ⟨I * J, I.isFractional.mul J.isFractional⟩ :=
by | Mathlib.RingTheory.FractionalIdeal.544_0.90B1BH8AtSmfl9S | theorem mul_def (I J : FractionalIdeal S P) : I * J = ⟨I * J, I.isFractional.mul J.isFractional⟩ | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : FractionalIdeal S P
⊢ ↑(I * J) = ↑I * ↑J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def, coe_mk] | @[simp, norm_cast]
theorem coe_mul (I J : FractionalIdeal S P) : (↑(I * J) : Submodule R P) = I * J := by
| Mathlib.RingTheory.FractionalIdeal.548_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem coe_mul (I J : FractionalIdeal S P) : (↑(I * J) : Submodule R P) = I * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Ideal R
⊢ ↑(I * J) = ↑I * ↑J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] | @[simp, norm_cast]
theorem coeIdeal_mul (I J : Ideal R) : (↑(I * J) : FractionalIdeal S P) = I * J := by
| Mathlib.RingTheory.FractionalIdeal.553_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem coeIdeal_mul (I J : Ideal R) : (↑(I * J) : FractionalIdeal S P) = I * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Ideal R
⊢ ↑(I * J) = { val := ↑↑I * ↑↑J, property := (_ : IsFractional S (↑↑I * ↑↑J)) } | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact coeToSubmodule_injective (coeSubmodule_mul _ _ _) | @[simp, norm_cast]
theorem coeIdeal_mul (I J : Ideal R) : (↑(I * J) : FractionalIdeal S P) = I * J := by
simp only [mul_def]
| Mathlib.RingTheory.FractionalIdeal.553_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem coeIdeal_mul (I J : Ideal R) : (↑(I * J) : FractionalIdeal S P) = I * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
⊢ Monotone fun x => I * x | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro J J' h | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) := by
| Mathlib.RingTheory.FractionalIdeal.559_0.90B1BH8AtSmfl9S | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J J' : FractionalIdeal S P
h : J ≤ J'
⊢ (fun x => I * x) J ≤ (fun x => I * x) J' | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) := by
intro J J' h
| Mathlib.RingTheory.FractionalIdeal.559_0.90B1BH8AtSmfl9S | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J J' : FractionalIdeal S P
h : J ≤ J'
⊢ { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } ≤
{ val := ↑I * ↑J', property := (_ : IsFractional S (↑I * ↑J')) } | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact mul_le.mpr fun x hx y hy => mul_mem_mul hx (h hy) | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) := by
intro J J' h
simp only [mul_def]
| Mathlib.RingTheory.FractionalIdeal.559_0.90B1BH8AtSmfl9S | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
⊢ Monotone fun J => J * I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro J J' h | theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J * I := by
| Mathlib.RingTheory.FractionalIdeal.565_0.90B1BH8AtSmfl9S | theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J * I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J J' : FractionalIdeal S P
h : J ≤ J'
⊢ (fun J => J * I) J ≤ (fun J => J * I) J' | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] | theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J * I := by
intro J J' h
| Mathlib.RingTheory.FractionalIdeal.565_0.90B1BH8AtSmfl9S | theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J * I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J J' : FractionalIdeal S P
h : J ≤ J'
⊢ { val := ↑J * ↑I, property := (_ : IsFractional S (↑J * ↑I)) } ≤
{ val := ↑J' * ↑I, property := (_ : IsFractional S (↑J' * ↑I)) } | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact mul_le.mpr fun x hx y hy => mul_mem_mul (h hx) hy | theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J * I := by
intro J J' h
simp only [mul_def]
| Mathlib.RingTheory.FractionalIdeal.565_0.90B1BH8AtSmfl9S | theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J * I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : FractionalIdeal S P
i j : P
hi : i ∈ I
hj : j ∈ J
⊢ i * j ∈ I * J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] | theorem mul_mem_mul {I J : FractionalIdeal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J := by
| Mathlib.RingTheory.FractionalIdeal.571_0.90B1BH8AtSmfl9S | theorem mul_mem_mul {I J : FractionalIdeal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : FractionalIdeal S P
i j : P
hi : i ∈ I
hj : j ∈ J
⊢ i * j ∈ { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact Submodule.mul_mem_mul hi hj | theorem mul_mem_mul {I J : FractionalIdeal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J := by
simp only [mul_def]
| Mathlib.RingTheory.FractionalIdeal.571_0.90B1BH8AtSmfl9S | theorem mul_mem_mul {I J : FractionalIdeal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J K : FractionalIdeal S P
⊢ I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] | theorem mul_le {I J K : FractionalIdeal S P} : I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K := by
| Mathlib.RingTheory.FractionalIdeal.577_0.90B1BH8AtSmfl9S | theorem mul_le {I J K : FractionalIdeal S P} : I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J K : FractionalIdeal S P
⊢ { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact Submodule.mul_le | theorem mul_le {I J K : FractionalIdeal S P} : I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K := by
simp only [mul_def]
| Mathlib.RingTheory.FractionalIdeal.577_0.90B1BH8AtSmfl9S | theorem mul_le {I J K : FractionalIdeal S P} : I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : FractionalIdeal S P
C : P → Prop
r : P
hr : r ∈ I * J
hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)
ha : ∀ (x y : P), C x → C y → C (x + y)
⊢ C r | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] at hr | @[elab_as_elim]
protected theorem mul_induction_on {I J : FractionalIdeal S P} {C : P → Prop} {r : P}
(hr : r ∈ I * J) (hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)) (ha : ∀ x y, C x → C y → C (x + y)) :
C r := by
| Mathlib.RingTheory.FractionalIdeal.590_0.90B1BH8AtSmfl9S | @[elab_as_elim]
protected theorem mul_induction_on {I J : FractionalIdeal S P} {C : P → Prop} {r : P}
(hr : r ∈ I * J) (hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)) (ha : ∀ x y, C x → C y → C (x + y)) :
C r | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : FractionalIdeal S P
C : P → Prop
r : P
hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)
ha : ∀ (x y : P), C x → C y → C (x + y)
hr : r ∈ { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) }
⊢ C... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact Submodule.mul_induction_on hr hm ha | @[elab_as_elim]
protected theorem mul_induction_on {I J : FractionalIdeal S P} {C : P → Prop} {r : P}
(hr : r ∈ I * J) (hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)) (ha : ∀ x y, C x → C y → C (x + y)) :
C r := by
simp only [mul_def] at hr
| Mathlib.RingTheory.FractionalIdeal.590_0.90B1BH8AtSmfl9S | @[elab_as_elim]
protected theorem mul_induction_on {I J : FractionalIdeal S P} {C : P → Prop} {r : P}
(hr : r ∈ I * J) (hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)) (ha : ∀ x y, C x → C y → C (x + y)) :
C r | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
n : ℕ
⊢ ↑(Nat.unaryCast n) = ↑n | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | induction n | theorem coe_nat_cast (n : ℕ) : ((n : FractionalIdeal S P) : Submodule R P) = n :=
show ((n.unaryCast : FractionalIdeal S P) : Submodule R P) = n
by | Mathlib.RingTheory.FractionalIdeal.601_0.90B1BH8AtSmfl9S | theorem coe_nat_cast (n : ℕ) : ((n : FractionalIdeal S P) : Submodule R P) = n | Mathlib_RingTheory_FractionalIdeal |
case zero
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
⊢ ↑(Nat.unaryCast Nat.zero) = ↑Nat.zero | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp [*, Nat.unaryCast] | theorem coe_nat_cast (n : ℕ) : ((n : FractionalIdeal S P) : Submodule R P) = n :=
show ((n.unaryCast : FractionalIdeal S P) : Submodule R P) = n
by induction n <;> | Mathlib.RingTheory.FractionalIdeal.601_0.90B1BH8AtSmfl9S | theorem coe_nat_cast (n : ℕ) : ((n : FractionalIdeal S P) : Submodule R P) = n | Mathlib_RingTheory_FractionalIdeal |
case succ
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
n✝ : ℕ
n_ih✝ : ↑(Nat.unaryCast n✝) = ↑n✝
⊢ ↑(Nat.unaryCast (Nat.succ n✝)) = ↑(Nat.succ n✝) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp [*, Nat.unaryCast] | theorem coe_nat_cast (n : ℕ) : ((n : FractionalIdeal S P) : Submodule R P) = n :=
show ((n.unaryCast : FractionalIdeal S P) : Submodule R P) = n
by induction n <;> | Mathlib.RingTheory.FractionalIdeal.601_0.90B1BH8AtSmfl9S | theorem coe_nat_cast (n : ℕ) : ((n : FractionalIdeal S P) : Submodule R P) = n | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
hI : 1 ≤ I
⊢ I ≤ I * I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | convert mul_left_mono I hI | theorem le_self_mul_self {I : FractionalIdeal S P} (hI : 1 ≤ I) : I ≤ I * I := by
| Mathlib.RingTheory.FractionalIdeal.636_0.90B1BH8AtSmfl9S | theorem le_self_mul_self {I : FractionalIdeal S P} (hI : 1 ≤ I) : I ≤ I * I | Mathlib_RingTheory_FractionalIdeal |
case h.e'_3
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
hI : 1 ≤ I
⊢ I = (fun x => I * x) 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact (mul_one I).symm | theorem le_self_mul_self {I : FractionalIdeal S P} (hI : 1 ≤ I) : I ≤ I * I := by
convert mul_left_mono I hI
| Mathlib.RingTheory.FractionalIdeal.636_0.90B1BH8AtSmfl9S | theorem le_self_mul_self {I : FractionalIdeal S P} (hI : 1 ≤ I) : I ≤ I * I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
hI : I ≤ 1
⊢ I * I ≤ I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | convert mul_left_mono I hI | theorem mul_self_le_self {I : FractionalIdeal S P} (hI : I ≤ 1) : I * I ≤ I := by
| Mathlib.RingTheory.FractionalIdeal.641_0.90B1BH8AtSmfl9S | theorem mul_self_le_self {I : FractionalIdeal S P} (hI : I ≤ 1) : I * I ≤ I | Mathlib_RingTheory_FractionalIdeal |
case h.e'_4
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
hI : I ≤ 1
⊢ I = (fun x => I * x) 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact (mul_one I).symm | theorem mul_self_le_self {I : FractionalIdeal S P} (hI : I ≤ 1) : I * I ≤ I := by
convert mul_left_mono I hI
| Mathlib.RingTheory.FractionalIdeal.641_0.90B1BH8AtSmfl9S | theorem mul_self_le_self {I : FractionalIdeal S P} (hI : I ≤ 1) : I * I ≤ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
⊢ J ≤ 1 ↔ ∃ I, ↑I = J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | constructor | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
| Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
⊢ J ≤ 1 → ∃ I, ↑I = J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro hJ | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ ∃ I, ↑I = J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩ | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
| Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ ∀ {a b : R},
a ∈ {x | (algebraMap R P) x ∈ J} → b ∈ {x | (algebraMap R P) x ∈ J} → a + b ∈ {x | (algebraMap R P) x ∈ J} | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro a b ha hb | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
a b : R
ha : a ∈ {x | (algebraMap R P) x ∈ J}
hb : b ∈ {x | (algebraMap R P) x ∈ J}
⊢ a + b ∈ {x | (algebraMap R P) x ∈ J} | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [mem_setOf, RingHom.map_add] | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
| Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
a b : R
ha : a ∈ {x | (algebraMap R P) x ∈ J}
hb : b ∈ {x | (algebraMap R P) x ∈ J}
⊢ (algebraMap R P) a + (algebraMap R P) b ∈ J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact J.val.add_mem ha hb | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
| Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_2
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ 0 ∈
{ carrier := {x | (algebraMap R P) x ∈ J},
add_mem' :=
(_ :
∀ {a b : R},
a ∈ {x | (alg... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [mem_setOf, RingHom.map_zero] | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_2
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ 0 ∈ J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact J.val.zero_mem | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_3
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ ∀ (c : R) {x : R},
x ∈
{
toAddSubsemigroup :=
{ carrier := {x | (algebraMap R P) x ∈ J},
... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro c x hx | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_3
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
c x : R
hx :
x ∈
{
toAddSubsemigroup :=
{ carrier := {x | (algebraMap R P) x ∈ J},
add_mem' :=
... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [smul_eq_mul, mem_setOf, RingHom.map_mul, ← Algebra.smul_def] | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_3
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
c x : R
hx :
x ∈
{
toAddSubsemigroup :=
{ carrier := {x | (algebraMap R P) x ∈ J},
add_mem' :=
... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact J.val.smul_mem c hx | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ ↑{
toAddSubmonoid :=
{
toAddSubsemigroup :=
{ carrier := {x | (algebraMap R P) x ∈ J},
... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | ext x | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4.a
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
⊢ x ∈
↑{
toAddSubmonoid :=
{
toAddSubsemigroup :=
{ carrier := {x | (algeb... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | constructor | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4.a.mp
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
⊢ x ∈
↑{
toAddSubmonoid :=
{
toAddSubsemigroup :=
{ carrier := {x | (al... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rintro ⟨y, hy, eq_y⟩ | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4.a.mp.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
y : R
hy :
y ∈
↑{
toAddSubmonoid :=
{
toAddSubsemigroup :=
{ carrie... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rwa [← eq_y] | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4.a.mpr
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
⊢ x ∈ J →
x ∈
↑{
toAddSubmonoid :=
{
toAddSubsemigroup :=
{ carrie... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro hx | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4.a.mpr
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
hx : x ∈ J
⊢ x ∈
↑{
toAddSubmonoid :=
{
toAddSubsemigroup :=
{ carrier := {x | (... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨y, rfl⟩ := (mem_one_iff S).mp (hJ hx) | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mp.refine'_4.a.mpr.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
hJ : J ≤ 1
y : R
hx : (algebraMap R P) y ∈ J
⊢ (algebraMap R P) y ∈
↑{
toAddSubmonoid :=
{
toAddSubsemigr... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact mem_setOf.mpr ⟨y, hx, rfl⟩ | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mpr
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
⊢ (∃ I, ↑I = J) → J ≤ 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rintro ⟨I, hI⟩ | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
I : Ideal R
hI : ↑I = J
⊢ J ≤ 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← hI] | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
J : FractionalIdeal S P
I : Ideal R
hI : ↑I = J
⊢ ↑I ≤ 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | apply coeIdeal_le_one | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J := by
constructor
· intro hJ
refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩
· intro a b ha hb
rw [mem_setOf, RingHom.map_add]
exact J.val.add_mem ha hb
· rw ... | Mathlib.RingTheory.FractionalIdeal.651_0.90B1BH8AtSmfl9S | theorem le_one_iff_exists_coeIdeal {J : FractionalIdeal S P} :
J ≤ (1 : FractionalIdeal S P) ↔ ∃ I : Ideal R, ↑I = J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : FractionalIdeal S P
⊢ 1 ≤ I ↔ 1 ∈ I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← coe_le_coe, coe_one, Submodule.one_le, mem_coe] | @[simp]
theorem one_le {I : FractionalIdeal S P} : 1 ≤ I ↔ (1 : P) ∈ I := by
| Mathlib.RingTheory.FractionalIdeal.676_0.90B1BH8AtSmfl9S | @[simp]
theorem one_le {I : FractionalIdeal S P} : 1 ≤ I ↔ (1 : P) ∈ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
⊢ ↑1 = 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [Ideal.one_eq_top, coeIdeal_top] | /-- `coeIdealHom (S : Submonoid R) P` is `(↑) : Ideal R → FractionalIdeal S P` as a ring hom -/
@[simps]
def coeIdealHom : Ideal R →+* FractionalIdeal S P where
toFun := coeIdeal
map_add' := coeIdeal_sup
map_mul' := coeIdeal_mul
map_one' := by | Mathlib.RingTheory.FractionalIdeal.683_0.90B1BH8AtSmfl9S | /-- `coeIdealHom (S : Submonoid R) P` is `(↑) : Ideal R → FractionalIdeal S P` as a ring hom -/
@[simps]
def coeIdealHom : Ideal R →+* FractionalIdeal S P where
toFun | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
g : P →ₐ[R] P'
I : S... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨b', b'_mem, hb'⟩ := Submodule.mem_map.mp hb | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
| Mathlib.RingTheory.FractionalIdeal.711_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ | Mathlib_RingTheory_FractionalIdeal |
case intro.intro
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
g :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [AlgHom.toLinearMap_apply] at hb' | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ := Submodule.mem_map.mp hb
| Mathlib.RingTheory.FractionalIdeal.711_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ | Mathlib_RingTheory_FractionalIdeal |
case intro.intro
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
g :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨x, hx⟩ := hI b' b'_mem | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ := Submodule.mem_map.mp hb
rw [AlgHom.toLinearMap_apply] at hb'
| Mathlib.RingTheory.FractionalIdeal.711_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | use x | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ := Submodule.mem_map.mp hb
rw [AlgHom.toLinearMap_apply] at hb'
obtain ⟨x, h... | Mathlib.RingTheory.FractionalIdeal.711_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ | Mathlib_RingTheory_FractionalIdeal |
case h
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
g : P →ₐ[R] P... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← g.commutes, hx, g.map_smul, hb'] | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ := Submodule.mem_map.mp hb
rw [AlgHom.toLinearMap_apply] at hb'
obtain ⟨x, h... | Mathlib.RingTheory.FractionalIdeal.711_0.90B1BH8AtSmfl9S | theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
| ⟨a, a_nonzero, hI⟩ =>
⟨a, a_nonzero, fun b hb => by
obtain ⟨b', b'_mem, hb'⟩ | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : FractionalIde... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | ext x | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
| Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
case a
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : Fracti... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mem_coeIdeal] | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
ext x
| Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
case a
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : Fracti... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | constructor | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
ext x
simp only [mem_coeIdeal]
| Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
case a.mp
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : Fra... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩ | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
ext x
simp only [mem_coeIdeal]
constructor
· | Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
case a.mp.intro.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocal... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact ⟨y, hy, (g.commutes y).symm⟩ | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
ext x
simp only [mem_coeIdeal]
constructor
· rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩
| Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
case a.mpr
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : Fr... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rintro ⟨y, hy, rfl⟩ | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
ext x
simp only [mem_coeIdeal]
constructor
· rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩
exact ⟨y, hy, (g.commutes y).symm⟩
· | Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
case a.mpr.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact ⟨_, ⟨y, hy, rfl⟩, g.commutes y⟩ | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by
ext x
simp only [mem_coeIdeal]
constructor
· rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩
exact ⟨y, hy, (g.commutes y).symm⟩
· rintro ⟨y, hy, rfl⟩
| Mathlib.RingTheory.FractionalIdeal.751_0.90B1BH8AtSmfl9S | @[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mul_def] | @[simp]
theorem map_mul : (I * J).map g = I.map g * J.map g := by
| Mathlib.RingTheory.FractionalIdeal.777_0.90B1BH8AtSmfl9S | @[simp]
theorem map_mul : (I * J).map g = I.map g * J.map g | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact coeToSubmodule_injective (Submodule.map_mul _ _ _) | @[simp]
theorem map_mul : (I * J).map g = I.map g * J.map g := by
simp only [mul_def]
| Mathlib.RingTheory.FractionalIdeal.777_0.90B1BH8AtSmfl9S | @[simp]
theorem map_mul : (I * J).map g = I.map g * J.map g | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← map_comp, g.symm_comp, map_id] | @[simp]
theorem map_map_symm (g : P ≃ₐ[R] P') : (I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I := by
| Mathlib.RingTheory.FractionalIdeal.783_0.90B1BH8AtSmfl9S | @[simp]
theorem map_map_symm (g : P ≃ₐ[R] P') : (I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : FractionalIde... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← map_comp, g.comp_symm, map_id] | @[simp]
theorem map_symm_map (I : FractionalIdeal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I := by
| Mathlib.RingTheory.FractionalIdeal.788_0.90B1BH8AtSmfl9S | @[simp]
theorem map_symm_map (I : FractionalIdeal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : FractionalIde... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← map_comp, AlgEquiv.symm_comp, map_id] | /-- If `g` is an equivalence, `map g` is an isomorphism -/
def mapEquiv (g : P ≃ₐ[R] P') : FractionalIdeal S P ≃+* FractionalIdeal S P' where
toFun := map g
invFun := map g.symm
map_add' I J := map_add I J _
map_mul' I J := map_mul I J _
left_inv I := by | Mathlib.RingTheory.FractionalIdeal.804_0.90B1BH8AtSmfl9S | /-- If `g` is an equivalence, `map g` is an isomorphism -/
def mapEquiv (g : P ≃ₐ[R] P') : FractionalIdeal S P ≃+* FractionalIdeal S P' where
toFun | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : FractionalIde... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [← map_comp, AlgEquiv.comp_symm, map_id] | /-- If `g` is an equivalence, `map g` is an isomorphism -/
def mapEquiv (g : P ≃ₐ[R] P') : FractionalIdeal S P ≃+* FractionalIdeal S P' where
toFun := map g
invFun := map g.symm
map_add' I J := map_add I J _
map_mul' I J := map_mul I J _
left_inv I := by rw [← map_comp, AlgEquiv.symm_comp, map_id]
right_inv... | Mathlib.RingTheory.FractionalIdeal.804_0.90B1BH8AtSmfl9S | /-- If `g` is an equivalence, `map g` is an isomorphism -/
def mapEquiv (g : P ≃ₐ[R] P') : FractionalIdeal S P ≃+* FractionalIdeal S P' where
toFun | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp | @[simp]
theorem mapEquiv_refl : mapEquiv AlgEquiv.refl = RingEquiv.refl (FractionalIdeal S P) :=
RingEquiv.ext fun x => by | Mathlib.RingTheory.FractionalIdeal.831_0.90B1BH8AtSmfl9S | @[simp]
theorem mapEquiv_refl : mapEquiv AlgEquiv.refl = RingEquiv.refl (FractionalIdeal S P) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [smul_zero] | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) :=
⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ =>
⟨a, a_mem, fun b hb =>
span_induction hb h
(by
| Mathlib.RingTheory.FractionalIdeal.836_0.90B1BH8AtSmfl9S | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact isInteger_zero | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) :=
⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ =>
⟨a, a_mem, fun b hb =>
span_induction hb h
(by
rw [smul_zero]
| Mathlib.RingTheory.FractionalIdeal.836_0.90B1BH8AtSmfl9S | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [smul_add] | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) :=
⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ =>
⟨a, a_mem, fun b hb =>
span_induction hb h
(by
rw [smul_zero]
exac... | Mathlib.RingTheory.FractionalIdeal.836_0.90B1BH8AtSmfl9S | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdea... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact isInteger_add hx hy | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) :=
⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ =>
⟨a, a_mem, fun b hb =>
span_induction hb h
(by
rw [smul_zero]
exac... | Mathlib.RingTheory.FractionalIdeal.836_0.90B1BH8AtSmfl9S | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) | Mathlib_RingTheory_FractionalIdeal |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.