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.a
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... | /-
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 | @[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 |
case neg.a
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... | /-
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... | dsimp only [val_eq_coe] at hx ⊢ | @[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 |
case neg.a
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... | /-
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_div h_spand, Submodule.mem_div_iff_forall_mul_mem] at hx | @[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 |
case neg.a
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... | /-
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... | specialize hx d (mem_spanSingleton_self R₁⁰ d) | @[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 |
case neg.a
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... | /-
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_xd : x = d⁻¹ * (x * d) := by field_simp | @[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 |
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_spand : sp... | /-
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... | field_simp | @[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 |
case neg.a
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... | /-
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_mul, one_div_spanSingleton, h_xd] | @[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 |
case neg.a
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... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact Submodule.mul_mem_mul (mem_spanSingleton_self R₁⁰ _) hx | @[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 |
case neg.a
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... | /-
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 [le_div_iff_mul_le h_spand, mul_assoc, mul_left_comm, one_div_spanSingleton,
spanSingleton_mul_spanSingleton, inv_mul_cancel hd, spanSingleton_one, mul_one] | @[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 |
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
I : FractionalIdeal R₁⁰ K
⊢ ∃ a aI, a ≠ 0 ∧ I = spanSing... | /-
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_inv, nonzero, ha⟩ := I.isFractional | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
| Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro
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
I : FractionalIdeal R₁⁰ K
a_inv : R₁
no... | /-
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 nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
| Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro
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
I : FractionalIdeal R₁⁰ K
a_inv : R₁
no... | /-
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 map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :=
mt IsFractionRing.to_map_eq_zero_iff.mp nonzero | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
| Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro
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
I : FractionalIdeal R₁⁰ K
a_inv : R₁
no... | /-
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_inv,
Submodule.comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (algebraMap R₁ K a_inv) * I),
nonzero, ext fun x => Iff.trans ⟨_, _⟩ mem_singleton_mul.symm⟩ | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_1
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
I : FractionalIdeal R₁⁰ K
a_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 hx | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_1
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
I : FractionalIdeal R₁⁰ K
a_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... | obtain ⟨x', hx'⟩ := ha x hx | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_1.intro
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
I : FractionalIdeal 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 [Algebra.smul_def] at hx' | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_1.intro
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
I : FractionalIdeal 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... | refine' ⟨algebraMap R₁ K x', (mem_coeIdeal _).mpr ⟨x', mem_singleton_mul.mpr _, rfl⟩, _⟩ | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_1.intro.refine'_1
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
I : Fractiona... | /-
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 ⟨x, hx, hx'⟩ | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_1.intro.refine'_2
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
I : Fractiona... | /-
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 [hx', ← mul_assoc, inv_mul_cancel map_a_nonzero, one_mul] | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_2
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
I : FractionalIdeal R₁⁰ K
a_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 ⟨y, hy, rfl⟩ | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_2.intro.intro
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
I : FractionalIde... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨x', hx', rfl⟩ := (mem_coeIdeal _).mp hy | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_2.intro.intro.intro.intro
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
I : F... | /-
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', hy', hx'⟩ := mem_singleton_mul.mp hx' | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_2.intro.intro.intro.intro.intro.intro
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✝ : IsDom... | /-
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.linearMap_apply] at hx' | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.refine'_2.intro.intro.intro.intro.intro.intro
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✝ : IsDom... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rwa [hx', ← mul_assoc, inv_mul_cancel map_a_nonzero, one_mul] | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI := by
obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional
have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero
have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :... | Mathlib.RingTheory.FractionalIdeal.1480_0.90B1BH8AtSmfl9S | theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI | Mathlib_RingTheory_FractionalIdeal |
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
inst✝³ :... | /-
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, aI, -, ha⟩ := exists_eq_spanSingleton_mul I | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
| Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
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 (algebraMap R K a)⁻¹ * algebraMap R K (generator aI) | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
| Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
case h
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
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... | suffices I = spanSingleton R⁰ ((algebraMap R K a)⁻¹ * algebraMap R K (generator aI)) by
rw [spanSingleton] at this
exact congr_arg Subtype.val this | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
| Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
inst✝³ :... | /-
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] at this | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
inst✝³ :... | /-
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 congr_arg Subtype.val this | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
case h
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
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... | conv_lhs => rw [ha, ← span_singleton_generator aI] | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
inst✝³ :... | /-
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, ← span_singleton_generator aI] | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
inst✝³ :... | /-
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, ← span_singleton_generator aI] | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
inst✝³ :... | /-
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, ← span_singleton_generator aI] | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | Mathlib_RingTheory_FractionalIdeal |
case h
R✝ : Type u_1
inst✝¹² : CommRing R✝
S : Submonoid R✝
P : Type u_2
inst✝¹¹ : CommRing P
inst✝¹⁰ : Algebra R✝ P
loc : IsLocalization S P
R₁ : Type u_3
inst✝⁹ : CommRing R₁
K : Type u_4
inst✝⁸ : Field K
inst✝⁷ : Algebra R₁ K
inst✝⁶ : IsFractionRing R₁ K
inst✝⁵ : IsDomain R₁
R : Type ?u.1581927
inst✝⁴ : CommRing R
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 [Ideal.submodule_span_eq, coeIdeal_span_singleton (generator aI),
spanSingleton_mul_spanSingleton] | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal := by
obtain ⟨a, aI, -, ha⟩ := exists_eq_spanSingleton_mul I
use (algebraMap R K a)⁻¹ * algebraMap R K (generator aI)
suffices I = spanSin... | Mathlib.RingTheory.FractionalIdeal.1503_0.90B1BH8AtSmfl9S | instance isPrincipal {R} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [Algebra R K]
[IsFractionRing R K] (I : FractionalIdeal R⁰ K) : (I : Submodule R K).IsPrincipal | 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 : P
I J : FractionalIdeal S P
⊢ (∀ {zI : P}, zI ∈ I → ... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [mem_singleton_mul, eq_comm] | theorem le_spanSingleton_mul_iff {x : P} {I J : FractionalIdeal S P} :
I ≤ spanSingleton S x * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI :=
show (∀ {zI} (hzI : zI ∈ I), zI ∈ spanSingleton _ x * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by
| Mathlib.RingTheory.FractionalIdeal.1515_0.90B1BH8AtSmfl9S | theorem le_spanSingleton_mul_iff {x : P} {I J : FractionalIdeal S P} :
I ≤ spanSingleton S x * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI | 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 : P
I J : FractionalIdeal S P
⊢ spanSingleton S 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... | simp only [mul_le, mem_singleton_mul, mem_spanSingleton] | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
| Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ 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 : P
I J : FractionalIdeal S P
⊢ (∀ (i : P), (∃ 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... | constructor | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
| Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_FractionalIdeal |
case mp
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 : P
I J : FractionalIdeal S P
⊢ (∀ (i : 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... | intro h zI hzI | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
· | Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_FractionalIdeal |
case mp
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 : P
I J : FractionalIdeal S P
h : ∀ (i : P), (... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact h x ⟨1, one_smul _ _⟩ zI hzI | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
· intro h zI hzI
| Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_FractionalIdeal |
case mpr
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 : P
I J : FractionalIdeal S P
⊢ (∀ z ∈ 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... | rintro h _ ⟨z, rfl⟩ zI hzI | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
· intro h zI hzI
exact h x ⟨1, one_smul _ _⟩ zI hzI
· | Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro
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 : P
I J : FractionalIdeal S P
h : ∀ 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 [Algebra.smul_mul_assoc] | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
· intro h zI hzI
exact h x ⟨1, one_smul _ _⟩ zI hzI
· rintro h _ ⟨z, rfl⟩ zI hzI
| Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | Mathlib_RingTheory_FractionalIdeal |
case mpr.intro
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 : P
I J : FractionalIdeal S P
h : ∀ 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... | exact Submodule.smul_mem J.1 _ (h zI hzI) | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
· intro h zI hzI
exact h x ⟨1, one_smul _ _⟩ zI hzI
· rintro h _ ⟨z, rfl⟩ zI hzI
rw [Algebra.smul_mul_assoc... | Mathlib.RingTheory.FractionalIdeal.1521_0.90B1BH8AtSmfl9S | theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ 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 : P
I J : FractionalIdeal S P
⊢ I = 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... | simp only [le_antisymm_iff, le_spanSingleton_mul_iff, spanSingleton_mul_le_iff] | theorem eq_spanSingleton_mul {x : P} {I J : FractionalIdeal S P} :
I = spanSingleton _ x * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by
| Mathlib.RingTheory.FractionalIdeal.1532_0.90B1BH8AtSmfl9S | theorem eq_spanSingleton_mul {x : P} {I J : FractionalIdeal S P} :
I = spanSingleton _ x * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁵ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝² : CommRing R₁
K : Type u_4
inst✝¹ : Field K
inst✝ : Algebra R₁ K
frac : IsFractionRing R₁ K
I : Submodule R₁ K
hI : I ≤ ↑0
⊢ FG 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_zero, le_bot_iff] at hI | theorem isNoetherian_zero : IsNoetherian R₁ (0 : FractionalIdeal R₁⁰ K) :=
isNoetherian_submodule.mpr fun I (hI : I ≤ (0 : FractionalIdeal R₁⁰ K)) => by
| Mathlib.RingTheory.FractionalIdeal.1545_0.90B1BH8AtSmfl9S | theorem isNoetherian_zero : IsNoetherian R₁ (0 : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁵ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝² : CommRing R₁
K : Type u_4
inst✝¹ : Field K
inst✝ : Algebra R₁ K
frac : IsFractionRing R₁ K
I : Submodule R₁ K
hI : I = ⊥
⊢ FG 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 [hI] | theorem isNoetherian_zero : IsNoetherian R₁ (0 : FractionalIdeal R₁⁰ K) :=
isNoetherian_submodule.mpr fun I (hI : I ≤ (0 : FractionalIdeal R₁⁰ K)) => by
rw [coe_zero, le_bot_iff] at hI
| Mathlib.RingTheory.FractionalIdeal.1545_0.90B1BH8AtSmfl9S | theorem isNoetherian_zero : IsNoetherian R₁ (0 : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝⁵ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
loc : IsLocalization S P
R₁ : Type u_3
inst✝² : CommRing R₁
K : Type u_4
inst✝¹ : Field K
inst✝ : Algebra R₁ K
frac : IsFractionRing R₁ K
I : Submodule R₁ K
hI : I = ⊥
⊢ FG ⊥ | /-
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 fg_bot | theorem isNoetherian_zero : IsNoetherian R₁ (0 : FractionalIdeal R₁⁰ K) :=
isNoetherian_submodule.mpr fun I (hI : I ≤ (0 : FractionalIdeal R₁⁰ K)) => by
rw [coe_zero, le_bot_iff] at hI
rw [hI]
| Mathlib.RingTheory.FractionalIdeal.1545_0.90B1BH8AtSmfl9S | theorem isNoetherian_zero : IsNoetherian R₁ (0 : 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
frac : IsFractionRing R₁ K
inst✝ : IsNoetherianRing R₁
I : Ideal R₁
⊢ IsNoetherian R₁ ↥↑↑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 [isNoetherian_iff] | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : FractionalIdeal R₁⁰ K) := by
| Mathlib.RingTheory.FractionalIdeal.1557_0.90B1BH8AtSmfl9S | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : 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
frac : IsFractionRing R₁ K
inst✝ : IsNoetherianRing R₁
I : Ideal R₁
⊢ ∀ J ≤ ↑I, FG ↑J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro J hJ | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : FractionalIdeal R₁⁰ K) := by
rw [isNoetherian_iff]
| Mathlib.RingTheory.FractionalIdeal.1557_0.90B1BH8AtSmfl9S | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : 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
frac : IsFractionRing R₁ K
inst✝ : IsNoetherianRing R₁
I : Ideal R₁
J : FractionalIdeal R₁⁰ K
hJ : J ≤ ↑I... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨J, rfl⟩ := le_one_iff_exists_coeIdeal.mp (le_trans hJ coeIdeal_le_one) | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : FractionalIdeal R₁⁰ K) := by
rw [isNoetherian_iff]
intro J hJ
| Mathlib.RingTheory.FractionalIdeal.1557_0.90B1BH8AtSmfl9S | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : 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
frac : IsFractionRing R₁ K
inst✝ : IsNoetherianRing R₁
I J : Ideal R₁
hJ : ↑J ≤ ↑I
⊢ FG ↑↑J | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact (IsNoetherian.noetherian J).map _ | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : FractionalIdeal R₁⁰ K) := by
rw [isNoetherian_iff]
intro J hJ
obtain ⟨J, rfl⟩ := le_one_iff_exists_coeIdeal.mp (le_trans hJ coeIdeal_le_one)
| Mathlib.RingTheory.FractionalIdeal.1557_0.90B1BH8AtSmfl9S | theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : 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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetherian R₁ ↥↑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... | by_cases hx : x = 0 | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
| Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case pos
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetheria... | /-
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 [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul] | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case pos
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetheria... | /-
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 isNoetherian_zero | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
| Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetheria... | /-
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_gx : algebraMap R₁ K x ≠ 0 :=
mt ((injective_iff_map_eq_zero (algebraMap R₁ K)).mp (IsFractionRing.injective _ _) x) hx | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetheria... | /-
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_spanx : spanSingleton R₁⁰ (algebraMap R₁ K x) ≠ 0 := spanSingleton_ne_zero_iff.mpr h_gx | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : IsNoetheria... | /-
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 [isNoetherian_iff] at hI ⊢ | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, FG... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro J hJ | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, FG... | /-
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_spanSingleton, le_div_iff_mul_le h_spanx] at hJ | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg
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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, FG... | /-
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 ⟨s, hs⟩ := hI _ hJ | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | Mathlib_RingTheory_FractionalIdeal |
case neg.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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ 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... | use s * {(algebraMap R₁ K x)⁻¹} | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | 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
frac : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, FG ↑... | /-
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 [Finset.coe_mul, Finset.coe_singleton, ← span_mul_span, hs, ← coe_spanSingleton R₁⁰, ←
coe_mul, mul_assoc, spanSingleton_mul_spanSingleton, mul_inv_cancel h_gx, spanSingleton_one,
mul_one] | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) := by
by_cases hx : x = 0
· rw [hx, RingHom.map_zero, inv_zero, spanSingleton_zero, zero_mul]
exact isNo... | Mathlib.RingTheory.FractionalIdeal.1567_0.90B1BH8AtSmfl9S | theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) | 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
frac : IsFractionRing R₁ K
inst✝¹ : IsDomain R₁
inst✝ : IsNoetherianRing R₁
I : FractionalIdeal R₁⁰ K
⊢ 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... | obtain ⟨d, J, _, rfl⟩ := exists_eq_spanSingleton_mul I | /-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I := by
| Mathlib.RingTheory.FractionalIdeal.1586_0.90B1BH8AtSmfl9S | /-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro
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
frac : IsFractionRing R₁ K
inst✝¹ : IsDomain R₁
inst✝ : IsNoetherianRing R₁
d : 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... | apply isNoetherian_spanSingleton_inv_to_map_mul | /-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I := by
obtain ⟨d, J, _, rfl⟩ := exists_eq_spanSingleton_mul I
| Mathlib.RingTheory.FractionalIdeal.1586_0.90B1BH8AtSmfl9S | /-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I | Mathlib_RingTheory_FractionalIdeal |
case intro.intro.intro.hI
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
frac : IsFractionRing R₁ K
inst✝¹ : IsDomain R₁
inst✝ : IsNoetherianRing R₁
d :... | /-
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 isNoetherian_coeIdeal | /-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I := by
obtain ⟨d, J, _, rfl⟩ := exists_eq_spanSingleton_mul I
apply isNoetherian_spanSingleton_inv_to_map_mul
| Mathlib.RingTheory.FractionalIdeal.1586_0.90B1BH8AtSmfl9S | /-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I | Mathlib_RingTheory_FractionalIdeal |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
⊢ (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have hxz := hxy.trans hyz | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxz : x < z
⊢ (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | rw [← sub_pos] at hxy hxz hyz | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
⊢ (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | suffices f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) by
ring_nf at this ⊢
linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
this : f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y)
⊢ (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | ring_nf at this ⊢ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
this : f y * (y - x)⁻¹ + f y * (-y + z)⁻¹ ≤ (y - x)⁻¹ * f x + (-y + z)⁻¹ * f z
⊢ f y * (y - x)⁻¹ - (y - x)⁻¹ * f x ≤ -(f y * (-y + z)⁻¹) + (-y + z)⁻¹ *... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
⊢ f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | set a := (z - y) / (z - x) | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
⊢ f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | set b := (y - x) / (z - x) | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
⊢ f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have hy : a • x + b • z = y := by field_simp; ring | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
⊢ a • x + b • z = y | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | field_simp | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
⊢ (z - y) * x + (y - x) * z = y * (z - x) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | ring | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have key :=
hf.2 hx hz (show 0 ≤ a by apply div_nonneg <;> linarith)
(show 0 ≤ b by apply div_nonneg <;> linarith)
(show a + b = 1 by field_simp) | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ 0 ≤ a | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | apply div_nonneg | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
case ha
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ 0 ≤ z - y | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
case hb
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ 0 ≤ z - x | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ 0 ≤ b | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | apply div_nonneg | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
case ha
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ 0 ≤ y - x | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
case hb
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ 0 ≤ z - x | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ a + b = 1 | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | field_simp | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
key : f (a • x + b • z) ≤ a • f x + b • f z
⊢ f y / (y - x) + f y / (z -... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | rw [hy] at key | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
key : f y ≤ a • f x + b • f z
⊢ f y / (y - x) + f y / (z - y) ≤ f x / (y... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | replace key := mul_le_mul_of_nonneg_left key hxz.le | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
key : (z - x) * f y ≤ (z - x) * (a • f x + b • f z)
⊢ f y / (y - x) + f ... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | field_simp [mul_comm (z - x) _] at key ⊢ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
key : f y * (z - x) ≤ (z - y) * f x + (y - x) * f z
⊢ (f y * (z - y) + f... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | rw [div_le_div_right] | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
key : f y * (z - x) ≤ (z - y) * f x + (y - x) * f z
⊢ f y * (z - y) + f ... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
key : f y * (z - x) ≤ (z - y) * f x + (y - x) * f z
⊢ 0 < (y - x) * (z -... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | nlinarith | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib.Analysis.Convex.Slope.24_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have := neg_le_neg (ConvexOn.slope_mono_adjacent hf.neg hx hz hxy hyz) | /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) ... | Mathlib.Analysis.Convex.Slope.48_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : -(((-f) z - (-f) y) / (z - y)) ≤ -(((-f) y - (-f) x) / (y - x))
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this | /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) ... | Mathlib.Analysis.Convex.Slope.48_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | exact this | /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) ... | Mathlib.Analysis.Convex.Slope.48_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have hxz := hxy.trans hyz | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxz : x < z
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have hxz' := hxz.ne | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxz : x < z
hxz' : x ≠ z
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | rw [← sub_pos] at hxy hxz hyz | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | suffices f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y) by
ring_nf at this ⊢
linarith | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
this : f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | ring_nf at this ⊢ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
this : f y * (y - x)⁻¹ + f y * (-y + z)⁻¹ < (y - x)⁻¹ * f x + (-y + z)⁻¹ * f z
⊢ f y * (y - x)⁻¹ - (y - x)⁻¹ * f x < -(f y * (-y + z... | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | linarith | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
⊢ f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | set a := (z - y) / (z - x) | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
a : 𝕜 := (z - y) / (z - x)
⊢ f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | set b := (y - x) / (z - x) | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
⊢ f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y) | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_import analysis.convex.slope from "leanp... | have hy : a • x + b • z = y := by field_simp; ring | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib.Analysis.Convex.Slope.57_0.2UqTeSfXEWgn9kZ | /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
(hx : x ∈ s) (hz : z ∈ ... | Mathlib_Analysis_Convex_Slope |
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