state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case neg
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
frac : IsFractionRing R₁ K
inst✝³ : IsDomain R₁
I✝ J✝ : FractionalIdeal R₁⁰ K
K' : Type u_5
inst... | /-
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 [Submodule.map_div] | @[simp]
theorem map_div (I J : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(I / J).map (h : K →ₐ[R₁] K') = I.map h / J.map h := by
by_cases H : J = 0
· rw [H, div_zero, map_zero, div_zero]
· -- Porting note: `simp` wouldn't apply these lemmas so do them manually using `rw`
rw [← coeToSubmodule_inj, div_non... | Mathlib.RingTheory.FractionalIdeal.1185_0.90B1BH8AtSmfl9S | @[simp]
theorem map_div (I J : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(I / J).map (h : K →ₐ[R₁] K') = I.map h / J.map h | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
frac : IsFractionRing R₁ K
inst✝³ : IsDomain R₁
I✝ J : FractionalIdeal R₁⁰ K
K' : Type u_5
inst✝² : Field... | /-
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_div, map_one] | theorem map_one_div (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(1 / I).map (h : K →ₐ[R₁] K') = 1 / I.map h := by | Mathlib.RingTheory.FractionalIdeal.1196_0.90B1BH8AtSmfl9S | theorem map_one_div (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(1 / I).map (h : K →ₐ[R₁] K') = 1 / I.map h | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractionRing 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... | rw [or_iff_not_imp_left] | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractionRing 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... | intro hI | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractionRing 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_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff] | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractionRing 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... | intro x | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractionRing 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... | constructor | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractio... | /-
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_mem | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· | Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFractio... | /-
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, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mp.intro.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝... | /-
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' ⟨n / d, _⟩ | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mp.intro.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝... | /-
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_div₀, IsFractionRing.mk'_eq_div] | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
refine' ⟨n / d, _⟩
| Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mpr
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : IsFracti... | /-
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 ⟨x, rfl⟩ | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
refine' ⟨n / d, _⟩
rw [map_div₀, IsFractionRing.m... | Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra K L
inst✝ : Is... | /-
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, y_ne, y_mem⟩ := exists_ne_zero_mem_isInteger hI | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
refine' ⟨n / d, _⟩
rw [map_div₀, IsFractionRing.m... | Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro.intro.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra 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... | rw [← div_mul_cancel x y_ne, RingHom.map_mul, ← Algebra.smul_def] | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
refine' ⟨n / d, _⟩
rw [map_div₀, IsFractionRing.m... | Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro.intro.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
K : Type u_4
L : Type u_5
inst✝⁶ : CommRing R₁
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra R₁ K
inst✝² : IsFractionRing R₁ K
inst✝¹ : Algebra 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 smul_mem (M := L) I (x / y) y_mem | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 := by
rw [or_iff_not_imp_left]
intro hI
simp_rw [@SetLike.ext_iff _ _ _ I 1, mem_one_iff]
intro x
constructor
· intro x_mem
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
refine' ⟨n / d, _⟩
rw [map_div₀, IsFractionRing.m... | Mathlib.RingTheory.FractionalIdeal.1208_0.90B1BH8AtSmfl9S | theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
⊢ IsFractional R₁⁰ (span R₁ (f '' ↑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 ⟨a', ha'⟩ := IsLocalization.exist_integer_multiples R₁⁰ s f | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ i ∈ s, IsInt... | /-
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' ⟨a', a'.2, fun x hx => Submodule.span_induction hx _ _ _ _⟩ | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_1
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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... | rintro _ ⟨i, hi, rfl⟩ | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_1.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁... | /-
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 ha' i hi | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_2
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 [smul_zero] | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_2
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 IsLocalization.isInteger_zero | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_3
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 y hx hy | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_3
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 [smul_add] | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_3
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 IsLocalization.isInteger_add hx hy | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_4
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 c x hx | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_4
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 [smul_comm] | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | Mathlib_RingTheory_FractionalIdeal |
case intro.refine'_4
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
a' : ↥R₁⁰
ha' : ∀ 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 IsLocalization.isInteger_smul hx | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R... | Mathlib.RingTheory.FractionalIdeal.1241_0.90B1BH8AtSmfl9S | /-- `FractionalIdeal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
-- Porting note: `@[simps]` generated a `Subtype.val` coercion instead of a
-- `FractionalIdeal.coeToSubmodule` coercion
def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
⊢ spanFinset R₁ s f = 0 ↔ ∀ j ∈ s, f 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 [← coeToSubmodule_inj, spanFinset_coe, coe_zero, Submodule.span_eq_bot,
Set.mem_image, Finset.mem_coe, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] | @[simp]
theorem spanFinset_eq_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
spanFinset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0 := by
| Mathlib.RingTheory.FractionalIdeal.1266_0.90B1BH8AtSmfl9S | @[simp]
theorem spanFinset_eq_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
spanFinset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
ι : Type u_5
s : Finset ι
f : ι → K
⊢ spanFinset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f 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 | theorem spanFinset_ne_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
spanFinset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0 := by | Mathlib.RingTheory.FractionalIdeal.1273_0.90B1BH8AtSmfl9S | theorem spanFinset_ne_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
spanFinset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
⊢ ↑(spanSingleton S x) = span R {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... | rw [spanSingleton] | @[simp]
theorem coe_spanSingleton (x : P) : (spanSingleton S x : Submodule R P) = span R {x} := by
| Mathlib.RingTheory.FractionalIdeal.1292_0.90B1BH8AtSmfl9S | @[simp]
theorem coe_spanSingleton (x : P) : (spanSingleton S x : Submodule R P) = span R {x} | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
⊢ ↑{ val := span R {x}, property := (_ : IsFractional S (span R {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... | rfl | @[simp]
theorem coe_spanSingleton (x : P) : (spanSingleton S x : Submodule R P) = span R {x} := by
rw [spanSingleton]
| Mathlib.RingTheory.FractionalIdeal.1292_0.90B1BH8AtSmfl9S | @[simp]
theorem coe_spanSingleton (x : P) : (spanSingleton S x : Submodule R P) = span R {x} | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
⊢ x ∈ spanSingleton S y ↔ ∃ z, z • y = 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... | rw [spanSingleton] | @[simp]
theorem mem_spanSingleton {x y : P} : x ∈ spanSingleton S y ↔ ∃ z : R, z • y = x := by
| Mathlib.RingTheory.FractionalIdeal.1298_0.90B1BH8AtSmfl9S | @[simp]
theorem mem_spanSingleton {x y : P} : x ∈ spanSingleton S y ↔ ∃ z : R, z • y = x | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
⊢ x ∈ { val := span R {y}, property := (_ : IsFractional S (span R {y... | /-
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.mem_span_singleton | @[simp]
theorem mem_spanSingleton {x y : P} : x ∈ spanSingleton S y ↔ ∃ z : R, z • y = x := by
rw [spanSingleton]
| Mathlib.RingTheory.FractionalIdeal.1298_0.90B1BH8AtSmfl9S | @[simp]
theorem mem_spanSingleton {x y : P} : x ∈ spanSingleton S y ↔ ∃ z : R, z • y = x | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
I : FractionalIdeal S P
⊢ spanSingleton S x ≤ I ↔ 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 [← coe_le_coe, coe_spanSingleton, Submodule.span_singleton_le_iff_mem, mem_coe] | @[simp]
theorem spanSingleton_le_iff_mem {x : P} {I : FractionalIdeal S P} :
spanSingleton S x ≤ I ↔ x ∈ I := by
| Mathlib.RingTheory.FractionalIdeal.1310_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_le_iff_mem {x : P} {I : FractionalIdeal S P} :
spanSingleton S x ≤ I ↔ x ∈ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : NoZeroSMulDivisors R P
x y : P
⊢ spanSingleton S x = spanSingleton 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... | rw [← Submodule.span_singleton_eq_span_singleton, spanSingleton, spanSingleton] | theorem spanSingleton_eq_spanSingleton [NoZeroSMulDivisors R P] {x y : P} :
spanSingleton S x = spanSingleton S y ↔ ∃ z : Rˣ, z • x = y := by
| Mathlib.RingTheory.FractionalIdeal.1316_0.90B1BH8AtSmfl9S | theorem spanSingleton_eq_spanSingleton [NoZeroSMulDivisors R P] {x y : P} :
spanSingleton S x = spanSingleton S y ↔ ∃ z : Rˣ, z • x = y | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : NoZeroSMulDivisors R P
x y : P
⊢ { val := span R {x}, property := (_... | /-
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 Subtype.mk_eq_mk | theorem spanSingleton_eq_spanSingleton [NoZeroSMulDivisors R P] {x y : P} :
spanSingleton S x = spanSingleton S y ↔ ∃ z : Rˣ, z • x = y := by
rw [← Submodule.span_singleton_eq_span_singleton, spanSingleton, spanSingleton]
| Mathlib.RingTheory.FractionalIdeal.1316_0.90B1BH8AtSmfl9S | theorem spanSingleton_eq_spanSingleton [NoZeroSMulDivisors R P] {x y : P} :
spanSingleton S x = spanSingleton S y ↔ ∃ z : Rˣ, z • x = y | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
I : FractionalIdeal S P
inst✝ : IsPrincipal ↑I
⊢ I = spanSingleton S (genera... | /-
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 [spanSingleton, ← coeToSubmodule_inj, coe_mk, span_singleton_generator] | theorem eq_spanSingleton_of_principal (I : FractionalIdeal S P) [IsPrincipal (I : Submodule R P)] :
I = spanSingleton S (generator (I : Submodule R P)) := by
-- Porting note: this used to be `coeToSubmodule_injective (span_singleton_generator ↑I).symm`
-- but Lean 4 struggled to unify everything. Turned it into... | Mathlib.RingTheory.FractionalIdeal.1322_0.90B1BH8AtSmfl9S | theorem eq_spanSingleton_of_principal (I : FractionalIdeal S P) [IsPrincipal (I : Submodule R P)] :
I = spanSingleton S (generator (I : Submodule R 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
⊢ spanSingleton S 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... | ext | @[simp]
theorem spanSingleton_zero : spanSingleton S (0 : P) = 0 := by
| Mathlib.RingTheory.FractionalIdeal.1335_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_zero : spanSingleton S (0 : P) = 0 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x✝ : P
⊢ x✝ ∈ spanSingleton S 0 ↔ 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... | simp [Submodule.mem_span_singleton, eq_comm] | @[simp]
theorem spanSingleton_zero : spanSingleton S (0 : P) = 0 := by
ext
| Mathlib.RingTheory.FractionalIdeal.1335_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_zero : spanSingleton S (0 : P) = 0 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
y : P
h : spanSingleton S y = 0
⊢ span R {y} = ⊥ | /-
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 using congr_arg Subtype.val h | theorem spanSingleton_eq_zero_iff {y : P} : spanSingleton S y = 0 ↔ y = 0 :=
⟨fun h =>
span_eq_bot.mp (by | Mathlib.RingTheory.FractionalIdeal.1341_0.90B1BH8AtSmfl9S | theorem spanSingleton_eq_zero_iff {y : P} : spanSingleton S y = 0 ↔ y = 0 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
y : P
h : y = 0
⊢ spanSingleton S y = 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 spanSingleton_eq_zero_iff {y : P} : spanSingleton S y = 0 ↔ y = 0 :=
⟨fun h =>
span_eq_bot.mp (by simpa using congr_arg Subtype.val h : span R {y} = ⊥) y (mem_singleton y),
fun h => by | Mathlib.RingTheory.FractionalIdeal.1341_0.90B1BH8AtSmfl9S | theorem spanSingleton_eq_zero_iff {y : P} : spanSingleton S y = 0 ↔ y = 0 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
⊢ spanSingleton S 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... | ext | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 := by
| Mathlib.RingTheory.FractionalIdeal.1351_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x✝ : P
⊢ x✝ ∈ spanSingleton S 1 ↔ 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... | refine' (mem_spanSingleton S).trans ((exists_congr _).trans (mem_one_iff S).symm) | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 := by
ext
| Mathlib.RingTheory.FractionalIdeal.1351_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x✝ : P
⊢ ∀ (a : R), a • 1 = x✝ ↔ (algebraMap R P) a = 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 x' | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 := by
ext
refine' (mem_spanSingleton S).trans ((exists_congr _).trans (mem_one_iff S).symm)
| Mathlib.RingTheory.FractionalIdeal.1351_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x✝ : P
x' : R
⊢ x' • 1 = x✝ ↔ (algebraMap R P) x' = 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... | rw [Algebra.smul_def, mul_one] | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 := by
ext
refine' (mem_spanSingleton S).trans ((exists_congr _).trans (mem_one_iff S).symm)
intro x'
| Mathlib.RingTheory.FractionalIdeal.1351_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
⊢ spanSingleton S x * spanSingleton S y = spanSingleton S (x * y) | /-
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 coeToSubmodule_injective | @[simp]
theorem spanSingleton_mul_spanSingleton (x y : P) :
spanSingleton S x * spanSingleton S y = spanSingleton S (x * y) := by
| Mathlib.RingTheory.FractionalIdeal.1359_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_mul_spanSingleton (x y : P) :
spanSingleton S x * spanSingleton S y = spanSingleton S (x * y) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
⊢ (fun I => ↑I) (spanSingleton S x * spanSingleton S y) = (fun... | /-
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 [coe_mul, coe_spanSingleton, span_mul_span, singleton_mul_singleton] | @[simp]
theorem spanSingleton_mul_spanSingleton (x y : P) :
spanSingleton S x * spanSingleton S y = spanSingleton S (x * y) := by
apply coeToSubmodule_injective
| Mathlib.RingTheory.FractionalIdeal.1359_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_mul_spanSingleton (x y : P) :
spanSingleton S x * spanSingleton S y = spanSingleton S (x * y) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
n : ℕ
⊢ spanSingleton S x ^ n = spanSingleton S (x ^ 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 with n hn | @[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) := by
| Mathlib.RingTheory.FractionalIdeal.1366_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) | Mathlib_RingTheory_FractionalIdeal |
case zero
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
⊢ spanSingleton S x ^ Nat.zero = spanSingleton S (x ^ Nat.zer... | /-
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 [pow_zero, pow_zero, spanSingleton_one] | @[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) := by
induction' n with n hn
· | Mathlib.RingTheory.FractionalIdeal.1366_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) | Mathlib_RingTheory_FractionalIdeal |
case succ
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
n : ℕ
hn : spanSingleton S x ^ n = spanSingleton S (x ^ 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... | rw [pow_succ, hn, spanSingleton_mul_spanSingleton, pow_succ] | @[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) := by
induction' n with n hn
· rw [pow_zero, pow_zero, spanSingleton_one]
· | Mathlib.RingTheory.FractionalIdeal.1366_0.90B1BH8AtSmfl9S | @[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : R
⊢ ↑(Ideal.span {x}) = spanSingleton S ((algebraMap R P) 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... | ext y | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
| Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : R
y : P
⊢ y ∈ ↑(Ideal.span {x}) ↔ y ∈ spanSingleton S ((algebraMap... | /-
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' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm) | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
| Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : R
y : P
⊢ (∃ x' ∈ Ideal.span {x}, (algebraMap R P) x' = y) ↔ ∃ z, ... | /-
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]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
| Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : R
y : P
⊢ (∃ x' ∈ Ideal.span {x}, (algebraMap R P) x' = y) → ∃ ... | /-
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]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· | Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | Mathlib_RingTheory_FractionalIdeal |
case a.mp.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y' : R
hy' : y' ∈ Ideal.span {x}
⊢ ∃ z, z • (algebraM... | /-
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', rfl⟩ := Submodule.mem_span_singleton.mp hy' | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· rintro ⟨y', hy', rfl⟩
| Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | Mathlib_RingTheory_FractionalIdeal |
case a.mp.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x x' : R
hy' : x' • x ∈ Ideal.span {x}
⊢ ∃ z, z •... | /-
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' | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· rintro ⟨y', hy', rfl⟩
obtain ⟨x', rfl⟩ := Submodule.mem_span_single... | Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x x' : R
hy' : x' • x ∈ Ideal.span {x}
⊢ x' • (algebraMap R P) 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... | rw [smul_eq_mul, RingHom.map_mul, Algebra.smul_def] | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· rintro ⟨y', hy', rfl⟩
obtain ⟨x', rfl⟩ := Submodule.mem_span_single... | Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : R
y : P
⊢ (∃ z, z • (algebraMap R P) x = y) → ∃ x' ∈ Ideal.spa... | /-
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', rfl⟩ | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· rintro ⟨y', hy', rfl⟩
obtain ⟨x', rfl⟩ := Submodule.mem_span_single... | Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | Mathlib_RingTheory_FractionalIdeal |
case a.mpr.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y' : R
⊢ ∃ x' ∈ Ideal.span {x}, (algebraMap R P) x' = y' •... | /-
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' ⟨y' * x, Submodule.mem_span_singleton.mpr ⟨y', rfl⟩, _⟩ | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· rintro ⟨y', hy', rfl⟩
obtain ⟨x', rfl⟩ := Submodule.mem_span_single... | Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | Mathlib_RingTheory_FractionalIdeal |
case a.mpr.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y' : R
⊢ (algebraMap R P) (y' * x) = y' • (algebraMap 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 [RingHom.map_mul, Algebra.smul_def] | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)
constructor
· rintro ⟨y', hy', rfl⟩
obtain ⟨x', rfl⟩ := Submodule.mem_span_single... | Mathlib.RingTheory.FractionalIdeal.1373_0.90B1BH8AtSmfl9S | @[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalizat... | /-
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 SetLike.ext_iff.mpr | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
| Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalizat... | /-
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 y | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalizat... | /-
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]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsL... | /-
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 h | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mpr
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : Is... | /-
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 h | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsL... | /-
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_spanSingleton] | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mp
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsL... | /-
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', rfl⟩ := (mem_canonicalEquiv_apply _ _ _).mp h | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mp.intro.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra 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... | obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp hx' | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mp.intro.intro.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebr... | /-
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 z | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case h
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLo... | /-
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 [IsLocalization.map_smul, RingHom.id_apply] | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mpr
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : Is... | /-
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_canonicalEquiv_apply] | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mpr
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : Is... | /-
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 ⟨z, rfl⟩ := (mem_spanSingleton _).mp h | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst... | /-
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 z • x | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case h
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLo... | /-
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 (mem_spanSingleton _).mpr ⟨z, rfl⟩ | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | Mathlib_RingTheory_FractionalIdeal |
case right
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁶ : CommRing R₁
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R₁ K
inst✝³ : IsFractionRing R₁ K
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : ... | /-
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 [IsLocalization.map_smul] | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) := by
apply SetLik... | Mathlib.RingTheory.FractionalIdeal.1388_0.90B1BH8AtSmfl9S | @[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
(x : P) :
canonicalEquiv S P P' (spanSingleton S x) =
spanSingleton S
(IsLocalization.map P' (RingHom.id R)
(fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
⊢ y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = 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... | constructor | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
| Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
⊢ y ∈ spanSingleton S x * I → ∃ y' ∈ ... | /-
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 h | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
h : y ∈ spanSingleton S x * I
⊢ ∃ y' ... | /-
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 FractionalIdeal.mul_induction_on h ?_ ?_ | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
| Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case mp.refine_1
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
h : y ∈ spanSingleton S 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 x' hx' y' hy' | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case mp.refine_1
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
h : y ∈ spanSingleton S 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 ⟨a, ha⟩ := (mem_spanSingleton S).mp hx' | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
| Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case mp.refine_1.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
h : y ∈ spanSingleton ... | /-
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 a • y', Submodule.smul_mem (I : Submodule R P) a hy' | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
obtain ⟨a, ha⟩ := (mem_spanSingleton S).mp hx'
| Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case right
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
h : y ∈ spanSingleton S 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... | rw [← ha, Algebra.mul_smul_comm, Algebra.smul_mul_assoc] | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
obtain ⟨a, ha⟩ := (mem_spanSingleton S).mp hx'
use a • y', Submodule.smul_mem (I ... | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case mp.refine_2
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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
h : y ∈ spanSingleton S 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... | rintro _ _ ⟨y, hy, rfl⟩ ⟨y', hy', rfl⟩ | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
obtain ⟨a, ha⟩ := (mem_spanSingleton S).mp hx'
use a • y', Submodule.smul_mem (I ... | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case mp.refine_2.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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y✝ : P
I : FractionalIdeal S P
h :... | /-
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 + y', Submodule.add_mem (I : Submodule R P) hy hy', (mul_add _ _ _).symm⟩ | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
obtain ⟨a, ha⟩ := (mem_spanSingleton S).mp hx'
use a • y', Submodule.smul_mem (I ... | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x y : P
I : FractionalIdeal S P
⊢ (∃ y' ∈ I, y = x * y') → y ∈ spanS... | /-
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⟩ | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
obtain ⟨a, ha⟩ := (mem_spanSingleton S).mp hx'
use a • y', Submodule.smul_mem (I ... | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | Mathlib_RingTheory_FractionalIdeal |
case 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
x : P
I : FractionalIdeal S P
y' : P
hy' : y' ∈ 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... | exact mul_mem_mul ((mem_spanSingleton S).mpr ⟨1, one_smul _ _⟩) hy' | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' := by
constructor
· intro h
refine FractionalIdeal.mul_induction_on h ?_ ?_
· intro x' hx' y' hy'
obtain ⟨a, ha⟩ := (mem_spanSingleton S).mp hx'
use a • y', Submodule.smul_mem (I ... | Mathlib.RingTheory.FractionalIdeal.1410_0.90B1BH8AtSmfl9S | theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
⊢ spanSingleton R₁⁰ (mk' K x { val := y,... | /-
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... | have :
spanSingleton R₁⁰ (IsLocalization.mk' _ (1 : R₁) ⟨y, hy⟩) *
spanSingleton R₁⁰ (algebraMap R₁ K y) =
1 := by
rw [spanSingleton_mul_spanSingleton, mul_comm, ← IsLocalization.mk'_eq_mul_mk'_one,
IsLocalization.mk'_self, spanSingleton_one] | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J := by
| Mathlib.RingTheory.FractionalIdeal.1427_0.90B1BH8AtSmfl9S | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
⊢ spanSingleton R₁⁰ (mk' K 1 { val := y,... | /-
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 [spanSingleton_mul_spanSingleton, mul_comm, ← IsLocalization.mk'_eq_mul_mk'_one,
IsLocalization.mk'_self, spanSingleton_one] | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J := by
have :
spanSingleton R₁⁰ (IsLocalization.mk' _ (1 : R₁) ⟨y, hy⟩) *
spanSingleton ... | Mathlib.RingTheory.FractionalIdeal.1427_0.90B1BH8AtSmfl9S | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
this : spanSingleton R₁⁰ (mk' K 1 { val ... | /-
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... | let y' : (FractionalIdeal R₁⁰ K)ˣ := Units.mkOfMulEqOne _ _ this | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J := by
have :
spanSingleton R₁⁰ (IsLocalization.mk' _ (1 : R₁) ⟨y, hy⟩) *
spanSingleton ... | Mathlib.RingTheory.FractionalIdeal.1427_0.90B1BH8AtSmfl9S | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
this : spanSingleton R₁⁰ (mk' K 1 { val ... | /-
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... | have coe_y' : ↑y' = spanSingleton R₁⁰ (IsLocalization.mk' K (1 : R₁) ⟨y, hy⟩) := rfl | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J := by
have :
spanSingleton R₁⁰ (IsLocalization.mk' _ (1 : R₁) ⟨y, hy⟩) *
spanSingleton ... | Mathlib.RingTheory.FractionalIdeal.1427_0.90B1BH8AtSmfl9S | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
this : spanSingleton R₁⁰ (mk' K 1 { val ... | /-
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' Iff.trans _ (y'.mul_right_inj.trans coeIdeal_inj) | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J := by
have :
spanSingleton R₁⁰ (IsLocalization.mk' _ (1 : R₁) ⟨y, hy⟩) *
spanSingleton ... | Mathlib.RingTheory.FractionalIdeal.1427_0.90B1BH8AtSmfl9S | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
this : spanSingleton R₁⁰ (mk' K 1 { val ... | /-
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_y', coeIdeal_mul, coeIdeal_span_singleton, coeIdeal_mul, coeIdeal_span_singleton, ←
mul_assoc, spanSingleton_mul_spanSingleton, ← mul_assoc, spanSingleton_mul_spanSingleton,
mul_comm (mk' _ _ _), ← IsLocalization.mk'_eq_mul_mk'_one, mul_comm (mk' _ _ _), ←
IsLocalization.mk'_eq_mul_mk'_one, IsLocali... | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J := by
have :
spanSingleton R₁⁰ (IsLocalization.mk' _ (1 : R₁) ⟨y, hy⟩) *
spanSingleton ... | Mathlib.RingTheory.FractionalIdeal.1427_0.90B1BH8AtSmfl9S | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
Ideal.span {x} * I = Ideal.span {y} * J | 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
R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
z : K
⊢ spanSingleton R₁⁰ z * ↑I = ↑J ↔ Ideal.span {(sec R₁⁰ z... | /-
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 [← mk'_mul_coeIdeal_eq_coeIdeal K (IsLocalization.sec R₁⁰ z).2.prop,
IsLocalization.mk'_sec K z] | theorem spanSingleton_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {z : K} :
spanSingleton R₁⁰ z * (I : FractionalIdeal R₁⁰ K) = J ↔
Ideal.span {((IsLocalization.sec R₁⁰ z).1 : R₁)} * I =
Ideal.span {((IsLocalization.sec R₁⁰ z).2 : R₁)} * J := by
| Mathlib.RingTheory.FractionalIdeal.1447_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {z : K} :
spanSingleton R₁⁰ z * (I : FractionalIdeal R₁⁰ K) = J ↔
Ideal.span {((IsLocalization.sec R₁⁰ z).1 : R₁)} * I =
Ideal.span {((IsLocalization.sec R₁⁰ z).2 : R₁)} * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : K
h : x = 0
⊢ 1 / spanSingleton R₁⁰ x = spanSingleto... | /-
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 one_div_spanSingleton (x : K) : 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹ :=
if h : x = 0 then by | Mathlib.RingTheory.FractionalIdeal.1457_0.90B1BH8AtSmfl9S | theorem one_div_spanSingleton (x : K) : 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹ | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : K
h : ¬x = 0
⊢ spanSingleton R₁⁰ x * spanSingleton 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 [h] | theorem one_div_spanSingleton (x : K) : 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹ :=
if h : x = 0 then by simp [h] else (eq_one_div_of_mul_eq_one_right _ _ (by | Mathlib.RingTheory.FractionalIdeal.1457_0.90B1BH8AtSmfl9S | theorem one_div_spanSingleton (x : K) : 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹ | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
⊢ J / spanSingleton R₁⁰ ... | /-
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 [← one_div_spanSingleton] | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J := by
| Mathlib.RingTheory.FractionalIdeal.1461_0.90B1BH8AtSmfl9S | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
⊢ J / spanSingleton R₁⁰ ... | /-
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... | by_cases hd : d = 0 | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J := by
rw [← one_div_spanSingleton]
| Mathlib.RingTheory.FractionalIdeal.1461_0.90B1BH8AtSmfl9S | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J | Mathlib_RingTheory_FractionalIdeal |
case pos
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : d = 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... | simp only [hd, spanSingleton_zero, div_zero, zero_mul] | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J := by
rw [← one_div_spanSingleton]
by_cases hd : d = 0
· | Mathlib.RingTheory.FractionalIdeal.1461_0.90B1BH8AtSmfl9S | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J | Mathlib_RingTheory_FractionalIdeal |
case neg
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 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... | have h_spand : spanSingleton R₁⁰ d ≠ 0 := mt spanSingleton_eq_zero_iff.mp hd | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J := by
rw [← one_div_spanSingleton]
by_cases hd : d = 0
· simp only [hd, spanSingleton_zero, div_zero, zero_mul]
| Mathlib.RingTheory.FractionalIdeal.1461_0.90B1BH8AtSmfl9S | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J | Mathlib_RingTheory_FractionalIdeal |
case neg
R : Type u_1
inst✝⁷ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁶ : CommRing P
inst✝⁵ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_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... | apply le_antisymm | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J := by
rw [← one_div_spanSingleton]
by_cases hd : d = 0
· simp only [hd, spanSingleton_zero, div_zero, zero_mul]
have h_spand : spanSingleton R₁⁰ d ≠ 0 := mt spanSingleton_eq_zero_iff.m... | Mathlib.RingTheory.FractionalIdeal.1461_0.90B1BH8AtSmfl9S | @[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J | Mathlib_RingTheory_FractionalIdeal |
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