state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝³ : Ring R
inst✝² : Ring S
inst✝¹ : IsDomain S
inst✝ : RingHomClass F R S
f : F
⊢ ker f ≠ ⊤ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Ne.def, Ideal.eq_top_iff_one] | /-- The kernel of a homomorphism to a domain is a prime ideal. -/
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S] [RingHomClass F R S] (f : F) :
(ker f).IsPrime :=
⟨by
| Mathlib.RingTheory.Ideal.Operations.2138_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a domain is a prime ideal. -/
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S] [RingHomClass F R S] (f : F) :
(ker f).IsPrime | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝³ : Ring R
inst✝² : Ring S
inst✝¹ : IsDomain S
inst✝ : RingHomClass F R S
f : F
⊢ 1 ∉ ker f | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact not_one_mem_ker f | /-- The kernel of a homomorphism to a domain is a prime ideal. -/
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S] [RingHomClass F R S] (f : F) :
(ker f).IsPrime :=
⟨by
rw [Ne.def, Ideal.eq_top_iff_one]
| Mathlib.RingTheory.Ideal.Operations.2138_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a domain is a prime ideal. -/
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S] [RingHomClass F R S] (f : F) :
(ker f).IsPrime | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝³ : Ring R
inst✝² : Ring S
inst✝¹ : IsDomain S
inst✝ : RingHomClass F R S
f : F
x y : R
⊢ x * y ∈ ker f → x ∈ ker f ∨ y ∈ ker f | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simpa only [mem_ker, map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _ | /-- The kernel of a homomorphism to a domain is a prime ideal. -/
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S] [RingHomClass F R S] (f : F) :
(ker f).IsPrime :=
⟨by
rw [Ne.def, Ideal.eq_top_iff_one]
exact not_one_mem_ker f,
fun {x y} => by
| Mathlib.RingTheory.Ideal.Operations.2138_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a domain is a prime ideal. -/
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S] [RingHomClass F R S] (f : F) :
(ker f).IsPrime | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
⊢ Ideal.IsMaximal (ker f) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker f).mp h1, fun J x hJ hxf hxJ => _⟩ | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
| Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
J : Ideal R
x : R
hJ : ker f ≤ J
hxf : x ∉ ker f
hxJ : x ∈ J
⊢ 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | obtain ⟨y, hy⟩ := hf (f x)⁻¹ | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker ... | Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
case intro
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
J : Ideal R
x : R
hJ : ker f ≤ J
hxf : x ∉ ker f
hxJ : x ∈ J
y : R
hy : f y = (f x)⁻¹
⊢ 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | have H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker ... | Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
case intro
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
J : Ideal R
x : R
hJ : ker f ≤ J
hxf : x ∉ ker f
hxJ : x ∈ J
y : R
hy : f y = (f x)⁻¹
H : 1 = y * x - (y * x - 1)
⊢ 1 ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [H] | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker ... | Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
case intro
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
J : Ideal R
x : R
hJ : ker f ≤ J
hxf : x ∉ ker f
hxJ : x ∈ J
y : R
hy : f y = (f x)⁻¹
H : 1 = y * x - (y * x - 1)
⊢ y * x - (y * x - 1) ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' J.sub_mem (J.mul_mem_left _ hxJ) (hJ _) | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker ... | Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
case intro
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
J : Ideal R
x : R
hJ : ker f ≤ J
hxf : x ∉ ker f
hxJ : x ∈ J
y : R
hy : f y = (f x)⁻¹
H : 1 = y * x - (y * x - 1)
⊢ y * x - 1 ∈ ker f | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [mem_ker] | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker ... | Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
case intro
R✝ : Type u
S : Type v
T : Type w
R : Type u_1
K : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Field K
inst✝ : RingHomClass F R K
f : F
hf : Function.Surjective ⇑f
J : Ideal R
x : R
hJ : ker f ≤ J
hxf : x ∉ ker f
hxJ : x ∈ J
y : R
hy : f y = (f x)⁻¹
H : 1 = y * x - (y * x - 1)
⊢ f (y * x - 1) = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [hy, map_sub, map_one, map_mul, inv_mul_cancel (mt (mem_ker f).mpr hxf), sub_self] | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal := by
refine'
Ideal.isMaximal_iff.mpr
⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker ... | Mathlib.RingTheory.Ideal.Operations.2148_0.5qK551sG47yBciY | /-- The kernel of a homomorphism to a field is a maximal ideal. -/
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K] [RingHomClass F R K] (f : F)
(hf : Function.Surjective f) : (ker f).IsMaximal | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Semiring R
inst✝ : Semiring S
rc : RingHomClass F R S
I : Ideal R
f : F
⊢ map f I = ⊥ ↔ I ≤ RingHom.ker f | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [RingHom.ker, eq_bot_iff, map_le_iff_le_comap] | theorem map_eq_bot_iff_le_ker {I : Ideal R} (f : F) : I.map f = ⊥ ↔ I ≤ RingHom.ker f := by
| Mathlib.RingTheory.Ideal.Operations.2172_0.5qK551sG47yBciY | theorem map_eq_bot_iff_le_ker {I : Ideal R} (f : F) : I.map f = ⊥ ↔ I ≤ RingHom.ker f | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝³ : Semiring R
inst✝² : Semiring S
rc : RingHomClass F R S
F' : Type u_4
inst✝¹ : RingEquivClass F' R S
f : F'
I : Ideal R
inst✝ : IsPrime I
⊢ IsPrime (map f I) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | have h : I.map f = I.map ((f : R ≃+* S) : R →+* S) := rfl | theorem map_isPrime_of_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') {I : Ideal R}
[IsPrime I] : IsPrime (map f I) := by
| Mathlib.RingTheory.Ideal.Operations.2180_0.5qK551sG47yBciY | theorem map_isPrime_of_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') {I : Ideal R}
[IsPrime I] : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝³ : Semiring R
inst✝² : Semiring S
rc : RingHomClass F R S
F' : Type u_4
inst✝¹ : RingEquivClass F' R S
f : F'
I : Ideal R
inst✝ : IsPrime I
h : map f I = map (↑↑f) I
⊢ IsPrime (map f I) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [h, map_comap_of_equiv I (f : R ≃+* S)] | theorem map_isPrime_of_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') {I : Ideal R}
[IsPrime I] : IsPrime (map f I) := by
have h : I.map f = I.map ((f : R ≃+* S) : R →+* S) := rfl
| Mathlib.RingTheory.Ideal.Operations.2180_0.5qK551sG47yBciY | theorem map_isPrime_of_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') {I : Ideal R}
[IsPrime I] : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝³ : Semiring R
inst✝² : Semiring S
rc : RingHomClass F R S
F' : Type u_4
inst✝¹ : RingEquivClass F' R S
f : F'
I : Ideal R
inst✝ : IsPrime I
h : map f I = map (↑↑f) I
⊢ IsPrime (comap (RingEquiv.symm ↑f) I) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact Ideal.IsPrime.comap (RingEquiv.symm (f : R ≃+* S)) | theorem map_isPrime_of_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') {I : Ideal R}
[IsPrime I] : IsPrime (map f I) := by
have h : I.map f = I.map ((f : R ≃+* S) : R →+* S) := rfl
rw [h, map_comap_of_equiv I (f : R ≃+* S)]
| Mathlib.RingTheory.Ideal.Operations.2180_0.5qK551sG47yBciY | theorem map_isPrime_of_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') {I : Ideal R}
[IsPrime I] : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
⊢ (∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' fun h => le_antisymm (le_sInf _) _ | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
| Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
⊢ ∀ b ∈ map f '' A, map f (sInf A) ≤ b | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro j hj y hy | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' A
y : S
hy : y ∈ map f (sInf A)
⊢ y ∈ j | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | cases' (mem_map_iff_of_surjective f hf).1 hy with x hx | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
| Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' A
y : S
hy : y ∈ map f (sInf A)
x : R
hx : x ∈ sInf A ∧ f x = y
⊢ y ∈ j | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | cases' (Set.mem_image _ _ _).mp hj with J hJ | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
| Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' A
y : S
hy : y ∈ map f (sInf A)
x : R
hx : x ∈ sInf A ∧ f x = y
J : Ideal R
hJ : J ∈ A... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← hJ.right, ← hx.right] | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' A
y : S
hy : y ∈ map f (sInf A)
x : R
hx : x ∈ sInf A ∧ f x = y
J : Ideal R
hJ : J ∈ A... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact mem_map_of_mem f (sInf_le_of_le hJ.left (le_of_eq rfl) hx.left) | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
⊢ sInf (map f '' A) ≤ map f (sInf A) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro y hy | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
⊢ y ∈ map f (sInf A) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | cases' hf y with x hx | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
⊢ y ∈ map f (sInf A) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' hx ▸ mem_map_of_mem f _ | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
⊢ x ∈ sInf A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | have : ∀ I ∈ A, y ∈ map f I := by simpa using hy | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
⊢ ∀ I ∈ A, y ∈ map f I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simpa using hy | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
this : ∀ I ∈ A, y ∈ map f I
⊢ x ∈ sInf A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [Submodule.mem_sInf] | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
this : ∀ I ∈ A, y ∈ map f I
⊢ ∀ p ∈ A, x ∈ p | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro J hJ | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
this : ∀ I ∈ A, y ∈ map f I
J : Submodule R R
hJ : J ∈ A
⊢ x ∈ J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rcases (mem_map_iff_of_surjective f hf).1 (this J hJ) with ⟨x', hx', rfl⟩ | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
x : R
J : Submodule R R
hJ : J ∈ A
x' : R
hx' : x' ∈ J
hy : f x' ∈ sInf (map f '' A)
hx : f x = f x'
this : ∀ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | have : x - x' ∈ J := by
apply h J hJ
rw [RingHom.mem_ker, map_sub, hx, sub_self] | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
x : R
J : Submodule R R
hJ : J ∈ A
x' : R
hx' : x' ∈ J
hy : f x' ∈ sInf (map f '' A)
hx : f x = f x'
this : ∀ I ∈ A, f x' ∈ map f I
⊢ x - x' ∈ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | apply h J hJ | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case a
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
x : R
J : Submodule R R
hJ : J ∈ A
x' : R
hx' : x' ∈ J
hy : f x' ∈ sInf (map f '' A)
hx : f x = f x'
this : ∀ I ∈ A, f x' ∈ map f I
⊢ x ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [RingHom.mem_ker, map_sub, hx, sub_self] | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
x : R
J : Submodule R R
hJ : J ∈ A
x' : R
hx' : x' ∈ J
hy : f x' ∈ sInf (map f '' A)
hx : f x = f x'
this✝ : ∀... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simpa only [sub_add_cancel] using J.add_mem this hx' | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
refine' fun h => le_antisymm (le_sInf _) _
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with ... | Mathlib.RingTheory.Ideal.Operations.2194_0.5qK551sG47yBciY | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
⊢ IsPrime (map f I) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩ | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
| Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
h : map f I = ⊤
⊢ ⊤ ≤ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | replace h := congr_arg (comap f) h | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
h : comap f (map f I) = comap f ⊤
⊢ ⊤ ≤ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [comap_map_of_surjective _ hf, comap_top] at h | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
| Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
h : I ⊔ comap f ⊥ = ⊤
⊢ ⊤ ≤ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact h ▸ sup_le (le_of_eq rfl) hk | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
| Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
⊢ x * y ∈ map f I → x ∈ map f I ∨ y ∈ map f I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' fun hxy => (hf x).recOn fun a ha => (hf y).recOn fun b hb => _ | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
hxy : x * y ∈ map f I
a : R
ha : f a = x
b : R
hb : f b = y
⊢ x ∈ map f I ∨ y ∈ map f I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← ha, ← hb, ← _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hxy : ∃ x ∈ I, f x = f (a * b)
hb : f b = y
⊢ x ∈ map f I ∨ y ∈ map f I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rcases hxy with ⟨c, hc, hc'⟩ | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hb : f b = y
c : R
hc : c ∈ I
hc' : f c = f (a * b)
⊢ x ∈ map f I ∨ y ∈ map f I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← sub_eq_zero, ← map_sub] at hc' | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hb : f b = y
c : R
hc : c ∈ I
hc' : f (c - a * b) = 0
⊢ x ∈ map f I ∨ y ∈ map f I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | have : a * b ∈ I := by
convert I.sub_mem hc (hk (hc' : c - a * b ∈ RingHom.ker f)) using 1
abel | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hb : f b = y
c : R
hc : c ∈ I
hc' : f (c - a * b) = 0
⊢ a * b ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | convert I.sub_mem hc (hk (hc' : c - a * b ∈ RingHom.ker f)) using 1 | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case h.e'_4
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hb : f b = y
c : R
hc : c ∈ I
hc' : f (c - a * b) = 0
⊢ a * b = c - (c - a * b) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | abel | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case h.e'_4
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hb : f b = y
c : R
hc : c ∈ I
hc' : f (c - a * b) = 0
⊢ a * b = c - (c - a * b) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | abel | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
hf : Function.Surjective ⇑f
I : Ideal R
H : IsPrime I
hk : RingHom.ker f ≤ I
x y : S
a : R
ha : f a = x
b : R
hb : f b = y
c : R
hc : c ∈ I
hc' : f (c - a * b) = 0
this : a * b ∈ I
⊢ x ∈ map f ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact
(H.mem_or_mem this).imp (fun h => ha ▸ mem_map_of_mem f h) fun h => hb ▸ mem_map_of_mem f h | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) := by
refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩
· replace h := congr_arg (comap f) h
rw [comap_map_of_surjective _ hf, comap_top] at h
exac... | Mathlib.RingTheory.Ideal.Operations.2215_0.5qK551sG47yBciY | theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
(hk : RingHom.ker f ≤ I) : IsPrime (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
I : Ideal R
f : F
hf : Function.Injective ⇑f
⊢ map f I = ⊥ ↔ I = ⊥ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [map_eq_bot_iff_le_ker, (RingHom.injective_iff_ker_eq_bot f).mp hf, le_bot_iff] | theorem map_eq_bot_iff_of_injective {I : Ideal R} {f : F} (hf : Function.Injective f) :
I.map f = ⊥ ↔ I = ⊥ := by
| Mathlib.RingTheory.Ideal.Operations.2232_0.5qK551sG47yBciY | theorem map_eq_bot_iff_of_injective {I : Ideal R} {f : F} (hf : Function.Injective f) :
I.map f = ⊥ ↔ I = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
I J : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
⊢ map f I = map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker f | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← (comap_injective_of_surjective f hf).eq_iff, comap_map_of_surjective f hf,
comap_map_of_surjective f hf, RingHom.ker_eq_comap_bot] | theorem map_eq_iff_sup_ker_eq_of_surjective {I J : Ideal R} (f : R →+* S)
(hf : Function.Surjective f) : map f I = map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker f := by
| Mathlib.RingTheory.Ideal.Operations.2244_0.5qK551sG47yBciY | theorem map_eq_iff_sup_ker_eq_of_surjective {I J : Ideal R} (f : R →+* S)
(hf : Function.Surjective f) : map f I = map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker f | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
⊢ map f (radical I) = radical (map f I) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [radical_eq_sInf, radical_eq_sInf] | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
| Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
⊢ map f (sInf {J | I ≤ J ∧ IsPrime J}) = sInf {J | map f I ≤ J ∧ IsPrime J} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
| Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
⊢ map f (sInf {J | I ≤ J ∧ IsPrime J}) = sInf {J | map f I ≤ J ∧ IsPrime J} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | convert map_sInf hf this | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
| Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
⊢ {J | map f I ≤ J ∧ IsPrime J} = map f '' {J | I ≤ J ∧ IsPrime J} | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' funext fun j => propext ⟨_, _⟩ | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3.refine'_1
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
j : Ideal S
⊢ setOf (fun J => map f I ≤ J ∧ IsPrime J) j → (map f '' {J | I ≤ J ∧ I... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rintro ⟨hj, hj'⟩ | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3.refine'_1.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
j : Ideal S
hj : map f I ≤ j
hj' : IsPrime j
⊢ (map f '' {J | I ≤ J ∧ IsPrime... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | haveI : j.IsPrime := hj' | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3.refine'_1.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this✝ : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
j : Ideal S
hj : map f I ≤ j
hj' this : IsPrime j
⊢ (map f '' {J | I ≤ J ∧ I... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact
⟨comap f j, ⟨⟨map_le_iff_le_comap.1 hj, comap_isPrime f j⟩, map_comap_of_surjective f hf j⟩⟩ | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3.refine'_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
j : Ideal S
⊢ (map f '' {J | I ≤ J ∧ IsPrime J}) j → setOf (fun J => map f I ≤ J ∧ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rintro ⟨J, ⟨hJ, hJ'⟩⟩ | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3.refine'_2.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
j : Ideal S
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ IsPrime J}
hJ' : map f J ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | haveI : J.IsPrime := hJ.right | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
case h.e'_3.h.e'_3.refine'_2.intro.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
I : Ideal R
h : RingHom.ker f ≤ I
this✝ : ∀ J ∈ {J | I ≤ J ∧ IsPrime J}, RingHom.ker f ≤ J
j : Ideal S
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ IsPrime J}
hJ' : map f J... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' ⟨hJ' ▸ map_mono hJ.left, hJ' ▸ map_isPrime_of_surjective hf (le_trans h hJ.left)⟩ | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf... | Mathlib.RingTheory.Ideal.Operations.2250_0.5qK551sG47yBciY | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ∀ (b : Submodule R M), 1 • b = b | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp | instance moduleSubmodule : Module (Ideal R) (Submodule R M) where
smul_add := smul_sup
add_smul := sup_smul
mul_smul := Submodule.smul_assoc
one_smul := by | Mathlib.RingTheory.Ideal.Operations.2276_0.5qK551sG47yBciY | instance moduleSubmodule : Module (Ideal R) (Submodule R M) where
smul_add | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
⊢ (fun b => g (f... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← map_one g, ← sub_eq_zero, ← map_sub g, ← mem_ker g] | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
⊢ f_inv 1 - 1 ∈ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | apply hg | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
case a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
⊢ f_inv 1... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [mem_ker f, map_sub f, sub_eq_zero, map_one f] | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
case a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
⊢ f (f_in... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact hf 1 | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
⊢ ∀ (x y : B),
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | intro x y | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
x y : B
⊢ OneHom... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← map_mul g, ← sub_eq_zero, ← map_sub g, ← mem_ker g] | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
x y : B
⊢ f_inv ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | apply hg | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
case a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
x y : B
⊢... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [mem_ker f, map_sub f, sub_eq_zero, map_mul f] | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
case a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
src✝ : B →+ C :=
(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }
x y : B
⊢... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [hf _] | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C :=
{ AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
toFun := fun b => g (f_inv ... | Mathlib.RingTheory.Ideal.Operations.2293_0.5qK551sG47yBciY | /-- Auxiliary definition used to define `liftOfRightInverse` -/
def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) :
B →+* C | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
φ : B →+* C
x : A
hx : x ∈ ker f
⊢ (comp φ f) x = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp [(mem_ker _).mp hx] | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib.RingTheory.Ideal.Operations.2319_0.5qK551sG47yBciY | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : { g // ker f ≤ ker g }
⊢ (fun φ => { val := comp φ f, property := (_ : ∀ x ∈ ker f, x ∈ ker (comp φ f)) })
((fun g => liftOfRightInverseAux f f_inv hf ↑g (_ : ker... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | ext | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib.RingTheory.Ideal.Operations.2319_0.5qK551sG47yBciY | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib_RingTheory_Ideal_Operations |
case a.a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : { g // ker f ≤ ker g }
x✝ : A
⊢ ↑((fun φ => { val := comp φ f, property := (_ : ∀ x ∈ ker f, x ∈ ker (comp φ f)) })
((fun g => liftOfRightInverseAux ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp only [comp_apply, liftOfRightInverseAux_comp_apply, Subtype.coe_mk] | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib.RingTheory.Ideal.Operations.2319_0.5qK551sG47yBciY | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
φ : B →+* C
⊢ (fun g => liftOfRightInverseAux f f_inv hf ↑g (_ : ker f ≤ ker ↑g))
((fun φ => { val := comp φ f, property := (_ : ∀ x ∈ ker f, x ∈ ker (comp φ f)) }) φ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | ext b | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib.RingTheory.Ideal.Operations.2319_0.5qK551sG47yBciY | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib_RingTheory_Ideal_Operations |
case a
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
φ : B →+* C
b : B
⊢ ((fun g => liftOfRightInverseAux f f_inv hf ↑g (_ : ker f ≤ ker ↑g))
((fun φ => { val := comp φ f, property := (_ : ∀ x ∈ ker f, x ∈ ker ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp [liftOfRightInverseAux, hf b] | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib.RingTheory.Ideal.Operations.2319_0.5qK551sG47yBciY | /-- `liftOfRightInverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`RingHom.liftOfRightInverse_comp`),
* where `f : A →+* B` has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
```
A .... | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
h : B →+* C
hh : comp h f = g
⊢ h = (liftOfRightInverse f f_inv hf) { val := g, property := hg } | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp_rw [← hh] | theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ := by
| Mathlib.RingTheory.Ideal.Operations.2369_0.5qK551sG47yBciY | theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ | Mathlib_RingTheory_Ideal_Operations |
A : Type u_1
B : Type u_2
C : Type u_3
inst✝² : Ring A
inst✝¹ : Ring B
inst✝ : Ring C
f : A →+* B
f_inv : B → A
hf : Function.RightInverse f_inv ⇑f
g : A →+* C
hg : ker f ≤ ker g
h : B →+* C
hh : comp h f = g
⊢ h = (liftOfRightInverse f f_inv hf) { val := comp h f, property := (_ : (fun g => ker f ≤ ker g) (comp h f)) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact ((f.liftOfRightInverse f_inv hf).apply_symm_apply _).symm | theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ := by
simp_rw [← hh]
| Mathlib.RingTheory.Ideal.Operations.2369_0.5qK551sG47yBciY | theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ | Mathlib_RingTheory_Ideal_Operations |
x : ℝ
n_large : 512 ≤ x
⊢ x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
| Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
⊢ x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
⊢ x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' ℝ) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_fo... | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
⊢ ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | intro x h5 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
x : ℝ
h5 : 0 < x
⊢ f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have h6 := mul_pos (zero_lt_two' ℝ) h5 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
x : ℝ
h5 : 0 < x
h6 : 0 < 2 * x
⊢ f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
x : ℝ
h5 : 0 < x
h6 : 0 < 2 * x
h7 : 0 < (2 * x) ^ sqrt (2 * x)
⊢ f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
⊢ x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
⊢ 0 < 512 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ←
mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3),
mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ 1 - 2 / 3 = 1 / 3 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ f x ≤ 0 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : ℝ) • LinearMap.id))) using 1
· e... | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) f | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | apply ConcaveOn.sub | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case hf
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) fun x => log x + sqrt (2 * x)... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | apply ConcaveOn.add | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case hf.hf
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) fun x => log x
case hf.hg
... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ 0 ≤ 0.5 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case hf.hg
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) fun x => sqrt (2 * x) * lo... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : ℝ) • LinearMap.id))) using 1 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case h.e'_9
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ Set.Ioi 0.5 = ⇑(2 • LinearMap.id) ⁻¹' Set.Ioi 1 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | ext x | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case h.e'_9.h
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x✝
x : ℝ
⊢ x ∈ Set.Ioi 0.5 ↔ x ∈ ⇑(2 • LinearMap.id... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul] | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case h.e'_9.h
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x✝
x : ℝ
⊢ OfScientific.ofScientific 5 true 1 < x ↔... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rw [← mul_lt_mul_left (two_pos)] | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case h.e'_9.h
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x✝
x : ℝ
⊢ 2 * OfScientific.ofScientific 5 true 1 <... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case h.e'_9.h
x✝ : ℝ
n_large : 512 ≤ x✝
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x✝
x : ℝ
⊢ 1 < 2 * x ↔ 1 < 2 * x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rfl | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case hg
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConvexOn ℝ (Set.Ioi 0.5) fun x => log 4 / 3 * x | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | apply ConvexOn.smul | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
case hg.hc
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ 0 ≤ log 4 / 3
case hg.hf
x : ℝ
n_large : 512 ≤ x
f :... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ 1 ≤ 4 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ 0 ≤ 3 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log... | Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc | /-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x | Mathlib_NumberTheory_Bertrand |
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