state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case refine'_1
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
r : ... | /-
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 [Algebra.smul_def] | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
r : ... | /-
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 mul_mem_right _ _ (mem_map_of_mem _ hr) | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
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... | rintro _ ⟨x, hx, rfl⟩ | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
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 [← mul_one (algebraMap R S x), ← Algebra.smul_def] | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_2.intro.intro
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
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 Submodule.smul_mem_smul hx Submodule.mem_top | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_3
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
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... | exact Submodule.zero_mem _ | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_4
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
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... | intro x y | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_4
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
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... | exact Submodule.add_mem _ | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
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... | intro a x hx | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Ideal R
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... | refine' Submodule.smul_induction_on hx _ _ | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5.refine'_1
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Id... | /-
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 r hr s _ | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5.refine'_1
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Id... | /-
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 [smul_comm] | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5.refine'_1
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Id... | /-
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 Submodule.smul_mem_smul hr Submodule.mem_top | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5.refine'_2
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Id... | /-
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 hx hy | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5.refine'_2
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Id... | /-
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 [smul_add] | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
case refine'_5.refine'_2
R✝ : Type u
S✝ : Type v
F : Type u_1
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S✝
rc : RingHomClass F R✝ S✝
f : F
I✝ J : Ideal R✝
K L : Ideal S✝
G : Type u_2
rcg : RingHomClass G S✝ R✝
ι : Sort u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
I : Id... | /-
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 Submodule.add_mem _ hx hy | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by
refine'
le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx =>
Submodule.span_induction hx _ _ _ _
· rw [A... | Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY | @[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
(I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : Semiring R
inst✝ : Semiring S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
G : Type u_2
rcg : RingHomClass G S R
ι : Sort u_3
hf : Function.Injective ⇑f
⊢ 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... | refine' le_trans (fun x hx => _) bot_le | theorem comap_bot_le_of_injective : comap f ⊥ ≤ I := by
| Mathlib.RingTheory.Ideal.Operations.1657_0.5qK551sG47yBciY | theorem comap_bot_le_of_injective : comap f ⊥ ≤ I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : Semiring R
inst✝ : Semiring S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
G : Type u_2
rcg : RingHomClass G S R
ι : Sort u_3
hf : Function.Injective ⇑f
x : R
hx : x ∈ comap f ⊥
⊢ 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 [mem_comap, Submodule.mem_bot, ← map_zero f] at hx | theorem comap_bot_le_of_injective : comap f ⊥ ≤ I := by
refine' le_trans (fun x hx => _) bot_le
| Mathlib.RingTheory.Ideal.Operations.1657_0.5qK551sG47yBciY | theorem comap_bot_le_of_injective : comap f ⊥ ≤ I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : Semiring R
inst✝ : Semiring S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
G : Type u_2
rcg : RingHomClass G S R
ι : Sort u_3
hf : Function.Injective ⇑f
x : R
hx : f x = f 0
⊢ 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... | exact Eq.symm (hf hx) ▸ Submodule.zero_mem ⊥ | theorem comap_bot_le_of_injective : comap f ⊥ ≤ I := by
refine' le_trans (fun x hx => _) bot_le
rw [mem_comap, Submodule.mem_bot, ← map_zero f] at hx
| Mathlib.RingTheory.Ideal.Operations.1657_0.5qK551sG47yBciY | theorem comap_bot_le_of_injective : comap f ⊥ ≤ I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : Semiring R
inst✝ : Semiring S
rc : RingHomClass F R S
f✝ : F
I✝ J : Ideal R
K L : Ideal S
G : Type u_2
rcg : RingHomClass G S R
ι : Sort u_3
I : Ideal R
f : R ≃+* S
⊢ map (↑(RingEquiv.symm 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 [← RingEquiv.toRingHom_eq_coe, ← RingEquiv.toRingHom_eq_coe, map_map,
RingEquiv.toRingHom_eq_coe, RingEquiv.toRingHom_eq_coe, RingEquiv.symm_comp, map_id] | /-- If `f : R ≃+* S` is a ring isomorphism and `I : Ideal R`, then `map f.symm (map f I) = I`. -/
@[simp]
theorem map_of_equiv (I : Ideal R) (f : R ≃+* S) :
(I.map (f : R →+* S)).map (f.symm : S →+* R) = I := by
| Mathlib.RingTheory.Ideal.Operations.1669_0.5qK551sG47yBciY | /-- If `f : R ≃+* S` is a ring isomorphism and `I : Ideal R`, then `map f.symm (map f I) = I`. -/
@[simp]
theorem map_of_equiv (I : Ideal R) (f : R ≃+* S) :
(I.map (f : R →+* S)).map (f.symm : S →+* R) = I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : Semiring R
inst✝ : Semiring S
rc : RingHomClass F R S
f✝ : F
I✝ J : Ideal R
K L : Ideal S
G : Type u_2
rcg : RingHomClass G S R
ι : Sort u_3
I : Ideal R
f : R ≃+* S
⊢ comap (↑f) (comap (↑(RingEquiv.symm 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 [← RingEquiv.toRingHom_eq_coe, ← RingEquiv.toRingHom_eq_coe, comap_comap,
RingEquiv.toRingHom_eq_coe, RingEquiv.toRingHom_eq_coe, RingEquiv.symm_comp, comap_id] | /-- If `f : R ≃+* S` is a ring isomorphism and `I : Ideal R`,
then `comap f (comap f.symm I) = I`. -/
@[simp]
theorem comap_of_equiv (I : Ideal R) (f : R ≃+* S) :
(I.comap (f.symm : S →+* R)).comap (f : R →+* S) = I := by
| Mathlib.RingTheory.Ideal.Operations.1677_0.5qK551sG47yBciY | /-- If `f : R ≃+* S` is a ring isomorphism and `I : Ideal R`,
then `comap f (comap f.symm I) = I`. -/
@[simp]
theorem comap_of_equiv (I : Ideal R) (f : R ≃+* S) :
(I.comap (f.symm : S →+* R)).comap (f : R →+* S) = I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Surjective ⇑f
I : Ideal R
r : R
h : f r ∈ map f I
s : R
hsi : s ∈ ↑I
hfsr : f s = f r
⊢ f (r - s) = 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... | rw [map_sub, hfsr, sub_self] | theorem comap_map_of_surjective (I : Ideal R) : comap f (map f I) = I ⊔ comap f ⊥ :=
le_antisymm
(fun r h =>
let ⟨s, hsi, hfsr⟩ := mem_image_of_mem_map_of_surjective f hf h
Submodule.mem_sup.2
⟨s, hsi, r - s, (Submodule.mem_bot S).2 <| by | Mathlib.RingTheory.Ideal.Operations.1716_0.5qK551sG47yBciY | theorem comap_map_of_surjective (I : Ideal R) : comap f (map f I) = I ⊔ comap f ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Surjective ⇑f
I : Ideal R
H : IsMaximal I
⊢ map f I = ⊤ ∨ IsMaximal (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' or_iff_not_imp_left.2 fun ne_top => ⟨⟨fun h => ne_top h, fun J hJ => _⟩⟩ | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) := by
| Mathlib.RingTheory.Ideal.Operations.1745_0.5qK551sG47yBciY | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Surjective ⇑f
I : Ideal R
H : IsMaximal I
ne_top : ¬map f I = ⊤
J : Ideal S
hJ : map f I < J
⊢ 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'
(relIsoOfSurjective f hf).injective
(Subtype.ext_iff.2 (Eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm)) | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) := by
refine' or_iff_not_imp_left.2 fun ne_top => ⟨⟨fun h => ne_top h, fun J hJ => _⟩⟩
· | Mathlib.RingTheory.Ideal.Operations.1745_0.5qK551sG47yBciY | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_1
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Surjective ⇑f
I : Ideal R
H : IsMaximal I
ne_top : ¬map f I = ⊤
J : Ideal S
hJ : map f I < J
⊢ I ≤ comap 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... | exact map_le_iff_le_comap.1 (le_of_lt hJ) | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) := by
refine' or_iff_not_imp_left.2 fun ne_top => ⟨⟨fun h => ne_top h, fun J hJ => _⟩⟩
· refine'
(relIsoOfSurjective f hf).injective
(Subtype.ext_iff.2 (Eq.trans (H.1.2 (comap f J) (lt... | Mathlib.RingTheory.Ideal.Operations.1745_0.5qK551sG47yBciY | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
case refine'_2
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Surjective ⇑f
I : Ideal R
H : IsMaximal I
ne_top : ¬map f I = ⊤
J : Ideal S
hJ : map f I < J
⊢ I ≠ comap 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... | exact fun h => hJ.right (le_map_of_comap_le_of_surjective f hf (le_of_eq h.symm)) | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) := by
refine' or_iff_not_imp_left.2 fun ne_top => ⟨⟨fun h => ne_top h, fun J hJ => _⟩⟩
· refine'
(relIsoOfSurjective f hf).injective
(Subtype.ext_iff.2 (Eq.trans (H.1.2 (comap f J) (lt... | Mathlib.RingTheory.Ideal.Operations.1745_0.5qK551sG47yBciY | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
⊢ IsMaximal (comap f K) | /-
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' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩ | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
| Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
⊢ 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... | suffices map f J = ⊤ by
have := congr_arg (comap f) this
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this
rw [eq_top_iff]
exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono bot_le) (le_of_lt hJ))) | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
| Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
this : map f J = ⊤
⊢ 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 := congr_arg (comap f) this | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
| Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
this✝ : map f J = ⊤
this : comap f (map f J) = comap 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... | rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
have := congr_arg (comap f) this
| Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
this✝ : map f J = ⊤
this : ⊤ ≤ J ⊔ comap 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... | rw [eq_top_iff] | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
have := congr_arg (comap f) this
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this
| Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
this✝ : map f J = ⊤
this : ⊤ ≤ J ⊔ comap 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... | exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono bot_le) (le_of_lt hJ))) | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
have := congr_arg (comap f) this
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this
rw [eq_top_iff]
| Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
⊢ 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'
H.1.2 (map f J)
(lt_of_le_of_ne (le_map_of_comap_le_of_surjective _ hf (le_of_lt hJ)) fun h =>
ne_of_lt hJ (_root_.trans (congr_arg (comap f) h) _)) | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
have := congr_arg (comap f) this
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this
rw [eq_top_iff]
exact le_trans... | Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
h : K = map f J
⊢ comap f (map f J) = 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 [comap_map_of_surjective _ hf, sup_eq_left] | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
have := congr_arg (comap f) this
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this
rw [eq_top_iff]
exact le_trans... | Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I : Ideal R
hf : Function.Surjective ⇑f
K : Ideal S
H : IsMaximal K
J : Ideal R
hJ : comap f K < J
h : K = map f J
⊢ comap 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... | exact le_trans (comap_mono bot_le) (le_of_lt hJ) | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) := by
refine' ⟨⟨comap_ne_top _ H.1.1, fun J hJ => _⟩⟩
suffices map f J = ⊤ by
have := congr_arg (comap f) this
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this
rw [eq_top_iff]
exact le_trans... | Mathlib.RingTheory.Ideal.Operations.1755_0.5qK551sG47yBciY | theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Bijective ⇑f
I : Ideal R
H : IsMaximal I
⊢ IsMaximal (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'
or_iff_not_imp_left.1 (map_eq_top_or_isMaximal_of_surjective f hf.right H) fun h => H.1.1 _ | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) := by
| Mathlib.RingTheory.Ideal.Operations.1798_0.5qK551sG47yBciY | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Bijective ⇑f
I : Ideal R
H : IsMaximal 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... | calc
I = comap f (map f I) := ((relIsoOfBijective f hf).right_inv I).symm
_ = comap f ⊤ := by rw [h]
_ = ⊤ := by rw [comap_top] | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) := by
refine'
or_iff_not_imp_left.1 (map_eq_top_or_isMaximal_of_surjective f hf.right H) fun h => H.1.1 _
| Mathlib.RingTheory.Ideal.Operations.1798_0.5qK551sG47yBciY | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Bijective ⇑f
I : Ideal R
H : IsMaximal I
h : map f I = ⊤
⊢ comap f (map f I) = comap 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 [h] | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) := by
refine'
or_iff_not_imp_left.1 (map_eq_top_or_isMaximal_of_surjective f hf.right H) fun h => H.1.1 _
calc
I = comap f (map f I) := ((relIsoOfBijective f hf).right_inv I).symm
_ = comap f ⊤ := by | Mathlib.RingTheory.Ideal.Operations.1798_0.5qK551sG47yBciY | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : RingHomClass F R S
f : F
I✝ : Ideal R
hf : Function.Bijective ⇑f
I : Ideal R
H : IsMaximal I
h : map f I = ⊤
⊢ comap 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_top] | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) := by
refine'
or_iff_not_imp_left.1 (map_eq_top_or_isMaximal_of_surjective f hf.right H) fun h => H.1.1 _
calc
I = comap f (map f I) := ((relIsoOfBijective f hf).right_inv I).symm
_ = comap f ⊤ := by rw [h]
_ = ⊤ := by | Mathlib.RingTheory.Ideal.Operations.1798_0.5qK551sG47yBciY | theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
r : R
hri : r ∈ I
s : R
hsj : s ∈ J
⊢ f (r * s) ∈ map f I * 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... | rw [_root_.map_mul] | theorem map_mul : map f (I * J) = map f I * map f J :=
le_antisymm
(map_le_iff_le_comap.2 <|
mul_le.2 fun r hri s hsj =>
show (f (r * s)) ∈ map f I * map f J by
| Mathlib.RingTheory.Ideal.Operations.1829_0.5qK551sG47yBciY | theorem map_mul : map f (I * J) = map f I * map f J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
r : R
hri : r ∈ I
s : R
hsj : s ∈ J
⊢ f r * f s ∈ map f I * 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... | exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj) | theorem map_mul : map f (I * J) = map f I * map f J :=
le_antisymm
(map_le_iff_le_comap.2 <|
mul_le.2 fun r hri s hsj =>
show (f (r * s)) ∈ map f I * map f J by
rw [_root_.map_mul]; | Mathlib.RingTheory.Ideal.Operations.1829_0.5qK551sG47yBciY | theorem map_mul : map f (I * J) = map f I * map f J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
i : S
x✝¹ : i ∈ ⇑f '' ↑I
r : R
hri : r ∈ ↑I
hfri : f r = i
j : S
x✝ : j ∈ ⇑f '' ↑J
s : R
hsj : s ∈ ↑J
hfsj : f s = j
⊢ f r * f s ∈ ↑(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... | rw [← _root_.map_mul] | theorem map_mul : map f (I * J) = map f I * map f J :=
le_antisymm
(map_le_iff_le_comap.2 <|
mul_le.2 fun r hri s hsj =>
show (f (r * s)) ∈ map f I * map f J by
rw [_root_.map_mul]; exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj))
(span_mul_span (↑f '' ↑I) (↑f '' ↑J) ▸ ... | Mathlib.RingTheory.Ideal.Operations.1829_0.5qK551sG47yBciY | theorem map_mul : map f (I * J) = map f I * map f J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
i : S
x✝¹ : i ∈ ⇑f '' ↑I
r : R
hri : r ∈ ↑I
hfri : f r = i
j : S
x✝ : j ∈ ⇑f '' ↑J
s : R
hsj : s ∈ ↑J
hfsj : f s = j
⊢ f (r * s) ∈ ↑(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... | exact mem_map_of_mem f (mul_mem_mul hri hsj) | theorem map_mul : map f (I * J) = map f I * map f J :=
le_antisymm
(map_le_iff_le_comap.2 <|
mul_le.2 fun r hri s hsj =>
show (f (r * s)) ∈ map f I * map f J by
rw [_root_.map_mul]; exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj))
(span_mul_span (↑f '' ↑I) (↑f '' ↑J) ▸ ... | Mathlib.RingTheory.Ideal.Operations.1829_0.5qK551sG47yBciY | theorem map_mul : map f (I * J) = map f I * map f J | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
⊢ ZeroHom.toFun { toFun := map f, map_zero' := (_ : map f ⊥ = ⊥) } 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... | simp only [one_eq_top] | /-- The pushforward `Ideal.map` as a monoid-with-zero homomorphism. -/
@[simps]
def mapHom : Ideal R →*₀ Ideal S where
toFun := map f
map_mul' I J := Ideal.map_mul f I J
map_one' := by | Mathlib.RingTheory.Ideal.Operations.1842_0.5qK551sG47yBciY | /-- The pushforward `Ideal.map` as a monoid-with-zero homomorphism. -/
@[simps]
def mapHom : Ideal R →*₀ Ideal S where
toFun | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
⊢ 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 Ideal.map_top f | /-- The pushforward `Ideal.map` as a monoid-with-zero homomorphism. -/
@[simps]
def mapHom : Ideal R →*₀ Ideal S where
toFun := map f
map_mul' I J := Ideal.map_mul f I J
map_one' := by simp only [one_eq_top]; | Mathlib.RingTheory.Ideal.Operations.1842_0.5qK551sG47yBciY | /-- The pushforward `Ideal.map` as a monoid-with-zero homomorphism. -/
@[simps]
def mapHom : Ideal R →*₀ Ideal S where
toFun | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
⊢ comap f (radical K) = radical (comap f K) | /-
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 | theorem comap_radical : comap f (radical K) = radical (comap f K) := by
| Mathlib.RingTheory.Ideal.Operations.1855_0.5qK551sG47yBciY | theorem comap_radical : comap f (radical K) = radical (comap f K) | Mathlib_RingTheory_Ideal_Operations |
case h
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
x✝ : R
⊢ x✝ ∈ comap f (radical K) ↔ x✝ ∈ radical (comap f K) | /-
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 [radical] | theorem comap_radical : comap f (radical K) = radical (comap f K) := by
ext
| Mathlib.RingTheory.Ideal.Operations.1855_0.5qK551sG47yBciY | theorem comap_radical : comap f (radical K) = radical (comap f K) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
hK : IsRadical K
⊢ IsRadical (Ideal.comap f K) | /-
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 [← hK.radical, comap_radical] | theorem IsRadical.comap (hK : K.IsRadical) : (comap f K).IsRadical := by
| Mathlib.RingTheory.Ideal.Operations.1862_0.5qK551sG47yBciY | theorem IsRadical.comap (hK : K.IsRadical) : (comap f K).IsRadical | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
hK : IsRadical K
⊢ IsRadical (Ideal.radical (Ideal.comap f K)) | /-
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 radical_isRadical | theorem IsRadical.comap (hK : K.IsRadical) : (comap f K).IsRadical := by
rw [← hK.radical, comap_radical]
| Mathlib.RingTheory.Ideal.Operations.1862_0.5qK551sG47yBciY | theorem IsRadical.comap (hK : K.IsRadical) : (comap f K).IsRadical | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
n : ℕ
⊢ comap f K ^ n ≤ comap f (K ^ n) | /-
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... | induction' n with n n_ih | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f := by
| Mathlib.RingTheory.Ideal.Operations.1879_0.5qK551sG47yBciY | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f | Mathlib_RingTheory_Ideal_Operations |
case zero
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
⊢ comap f K ^ Nat.zero ≤ comap f (K ^ Nat.zero) | /-
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 [pow_zero, pow_zero, Ideal.one_eq_top, Ideal.one_eq_top] | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f := by
induction' n with n n_ih
· | Mathlib.RingTheory.Ideal.Operations.1879_0.5qK551sG47yBciY | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f | Mathlib_RingTheory_Ideal_Operations |
case zero
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
⊢ ⊤ ≤ comap 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 rfl.le | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f := by
induction' n with n n_ih
· rw [pow_zero, pow_zero, Ideal.one_eq_top, Ideal.one_eq_top]
| Mathlib.RingTheory.Ideal.Operations.1879_0.5qK551sG47yBciY | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f | Mathlib_RingTheory_Ideal_Operations |
case succ
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
n : ℕ
n_ih : comap f K ^ n ≤ comap f (K ^ n)
⊢ comap f K ^ Nat.succ n ≤ comap f (K ^ Nat.succ n) | /-
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 [pow_succ, pow_succ] | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f := by
induction' n with n n_ih
· rw [pow_zero, pow_zero, Ideal.one_eq_top, Ideal.one_eq_top]
exact rfl.le
· | Mathlib.RingTheory.Ideal.Operations.1879_0.5qK551sG47yBciY | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f | Mathlib_RingTheory_Ideal_Operations |
case succ
R : Type u
S : Type v
F : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
rc : RingHomClass F R S
f : F
I J : Ideal R
K L : Ideal S
n : ℕ
n_ih : comap f K ^ n ≤ comap f (K ^ n)
⊢ comap f K * comap f K ^ n ≤ comap f (K * K ^ n) | /-
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.mul_mono_right n_ih).trans (Ideal.le_comap_mul f) | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f := by
induction' n with n n_ih
· rw [pow_zero, pow_zero, Ideal.one_eq_top, Ideal.one_eq_top]
exact rfl.le
· rw [pow_succ, pow_succ]
| Mathlib.RingTheory.Ideal.Operations.1879_0.5qK551sG47yBciY | theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f | Mathlib_RingTheory_Ideal_Operations |
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hi : IsPrimary I
x y : R
x✝ : x * y ∈ radical I
m : ℕ
hxy : (x * y) ^ m ∈ I
⊢ x ∈ radical I ∨ y ∈ radical 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 [mul_pow] at hxy | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) :=
⟨mt radical_eq_top.1 hi.1,
fun {x y} ⟨m, hxy⟩ => by
| Mathlib.RingTheory.Ideal.Operations.1909_0.5qK551sG47yBciY | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hi : IsPrimary I
x y : R
x✝ : x * y ∈ radical I
m : ℕ
hxy : x ^ m * y ^ m ∈ I
⊢ x ∈ radical I ∨ y ∈ radical 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... | cases' hi.2 hxy with h h | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) :=
⟨mt radical_eq_top.1 hi.1,
fun {x y} ⟨m, hxy⟩ => by
rw [mul_pow] at hxy; | Mathlib.RingTheory.Ideal.Operations.1909_0.5qK551sG47yBciY | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) | Mathlib_RingTheory_Ideal_Operations |
case inl
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hi : IsPrimary I
x y : R
x✝ : x * y ∈ radical I
m : ℕ
hxy : x ^ m * y ^ m ∈ I
h : x ^ m ∈ I
⊢ x ∈ radical I ∨ y ∈ radical 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 Or.inl ⟨m, h⟩ | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) :=
⟨mt radical_eq_top.1 hi.1,
fun {x y} ⟨m, hxy⟩ => by
rw [mul_pow] at hxy; cases' hi.2 hxy with h h
· | Mathlib.RingTheory.Ideal.Operations.1909_0.5qK551sG47yBciY | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) | Mathlib_RingTheory_Ideal_Operations |
case inr
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hi : IsPrimary I
x y : R
x✝ : x * y ∈ radical I
m : ℕ
hxy : x ^ m * y ^ m ∈ I
h : y ^ m ∈ radical I
⊢ x ∈ radical I ∨ y ∈ radical 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 Or.inr (mem_radical_of_pow_mem h) | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) :=
⟨mt radical_eq_top.1 hi.1,
fun {x y} ⟨m, hxy⟩ => by
rw [mul_pow] at hxy; cases' hi.2 hxy with h h
· exact Or.inl ⟨m, h⟩
· | Mathlib.RingTheory.Ideal.Operations.1909_0.5qK551sG47yBciY | theorem isPrime_radical {I : Ideal R} (hi : IsPrimary I) : IsPrime (radical I) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
⊢ x ∈ I ⊓ J ∨ y ∈ radical (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... | rw [radical_inf, hij, inf_idem] | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
| Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
⊢ x ∈ I ⊓ J ∨ y ∈ radical 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' hi.2 hxyi with hxi hyi | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
rw [radical_inf, hij, inf_idem]
| Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
case inl
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
hxi : x ∈ I
⊢ x ∈ I ⊓ J ∨ y ∈ radical J
case inr
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : r... | /-
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' hj.2 hxyj with hxj hyj | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
rw [radical_inf, hij, inf_idem]
cases' hi.2 hxyi with hxi hyi; | Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
case inl.inl
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
hxi : x ∈ I
hxj : x ∈ J
⊢ x ∈ I ⊓ J ∨ y ∈ radical 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... | exact Or.inl ⟨hxi, hxj⟩ | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
rw [radical_inf, hij, inf_idem]
cases' hi.2 hxyi with hxi hyi; cases' hj.2 hx... | Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
case inl.inr
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
hxi : x ∈ I
hyj : y ∈ radical J
⊢ x ∈ I ⊓ J ∨ y ∈ radical 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... | exact Or.inr hyj | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
rw [radical_inf, hij, inf_idem]
cases' hi.2 hxyi with hxi hyi; cases' hj.2 hx... | Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
case inr
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
hyi : y ∈ radical I
⊢ x ∈ I ⊓ J ∨ y ∈ radical 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 [hij] at hyi | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
rw [radical_inf, hij, inf_idem]
cases' hi.2 hxyi with hxi hyi; cases' hj.2 hx... | Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
case inr
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
hi : IsPrimary I
hj : IsPrimary J
hij : radical I = radical J
x y : R
x✝ : x * y ∈ I ⊓ J
hxyi : x * y ∈ ↑I
hxyj : x * y ∈ ↑J
hyi : y ∈ radical J
⊢ x ∈ I ⊓ J ∨ y ∈ radical 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... | exact Or.inr hyi | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) :=
⟨ne_of_lt <| lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1),
fun {x y} ⟨hxyi, hxyj⟩ => by
rw [radical_inf, hij, inf_idem]
cases' hi.2 hxyi with hxi hyi; cases' hj.2 hx... | Mathlib.RingTheory.Ideal.Operations.1917_0.5qK551sG47yBciY | theorem isPrimary_inf {I J : Ideal R} (hi : IsPrimary I) (hj : IsPrimary J)
(hij : radical I = radical J) : IsPrimary (I ⊓ J) | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
f : ι →₀ ↥I
⊢ (finsuppTotal ι M I v) f = Finsupp.sum f fun i x => ↑x • v 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... | dsimp [finsuppTotal] | theorem finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i := by
| Mathlib.RingTheory.Ideal.Operations.1948_0.5qK551sG47yBciY | theorem finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
f : ι →₀ ↥I
⊢ (Finsupp.total ι M R v) (Finsupp.mapRange Subtype.val (_ : (Submodule.subtype I) 0 = 0) f) =
Finsupp.sum f fun i x => ↑x • v 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 [Finsupp.total_apply, Finsupp.sum_mapRange_index] | theorem finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i := by
dsimp [finsuppTotal]
| Mathlib.RingTheory.Ideal.Operations.1948_0.5qK551sG47yBciY | theorem finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
f : ι →₀ ↥I
⊢ ∀ (a : ι), 0 • v a = 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... | exact fun _ => zero_smul _ _ | theorem finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i := by
dsimp [finsuppTotal]
rw [Finsupp.total_apply, Finsupp.sum_mapRange_index]
| Mathlib.RingTheory.Ideal.Operations.1948_0.5qK551sG47yBciY | theorem finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
inst✝ : Fintype ι
f : ι →₀ ↥I
⊢ (finsuppTotal ι M I v) f = ∑ i : ι, ↑(f i) • v 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 [finsuppTotal_apply, Finsupp.sum_fintype] | theorem finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
finsuppTotal ι M I v f = ∑ i, (f i : R) • v i := by
| Mathlib.RingTheory.Ideal.Operations.1955_0.5qK551sG47yBciY | theorem finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
finsuppTotal ι M I v f = ∑ i, (f i : R) • v i | Mathlib_RingTheory_Ideal_Operations |
case h
ι : Type u_1
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
inst✝ : Fintype ι
f : ι →₀ ↥I
⊢ ∀ (i : ι), ↑0 • v i = 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... | exact fun _ => zero_smul _ _ | theorem finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
finsuppTotal ι M I v f = ∑ i, (f i : R) • v i := by
rw [finsuppTotal_apply, Finsupp.sum_fintype]
| Mathlib.RingTheory.Ideal.Operations.1955_0.5qK551sG47yBciY | theorem finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
finsuppTotal ι M I v f = ∑ i, (f i : R) • v i | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
⊢ LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | /-
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 | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
| Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
x✝ : M
⊢ x✝ ∈ LinearMap.range (finsuppTotal ι M I v) ↔ x✝ ∈ I • Submodule.span R (Set.range v) | /-
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_ideal_smul_span_iff_exists_sum] | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
| Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
x✝ : M
⊢ x✝ ∈ LinearMap.range (finsuppTotal ι M I v) ↔ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • v 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... | refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩ | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
| Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
x✝ : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • v i) = x✝) →
x✝ ∈ LinearMap.range (finsuppTotal ι M I v) | /-
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 ⟨a, ha, rfl⟩ | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
| Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.sum a fun i c => c • v i) ∈ LinearMap.range (finsuppTotal ι M I v) | /-
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... | classical
refine' ⟨a.mapRange (fun r => if h : r ∈ I then ⟨r, h⟩ else 0) (by simp), _⟩
rw [finsuppTotal_apply, Finsupp.sum_mapRange_index]
· apply Finsupp.sum_congr
intro i _
rw [dif_pos (ha i)]
· exact fun _ => zero_smul _ _ | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
| Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.sum a fun i c => c • v i) ∈ LinearMap.range (finsuppTotal ι M I v) | /-
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' ⟨a.mapRange (fun r => if h : r ∈ I then ⟨r, h⟩ else 0) (by simp), _⟩ | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
| Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (fun r => if h : r ∈ I then { val := r, property := h } else 0) 0 = 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 | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
refine' ⟨a.mapRang... | Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (finsuppTotal ι M I v)
(Finsupp.mapRange (fun r => if h : r ∈ I then { val := r, property := h... | /-
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 [finsuppTotal_apply, Finsupp.sum_mapRange_index] | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
refine' ⟨a.mapRang... | Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.sum a fun a b => ↑(if h : b ∈ I then { val := b, property := h } else 0) • v a) =
Finsu... | /-
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 Finsupp.sum_congr | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
refine' ⟨a.mapRang... | Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro.h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ ∀ x ∈ a.support, ↑(if h : a x ∈ I then { val := a x, property := h } else 0) • v x = a x • v 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... | intro i _ | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
refine' ⟨a.mapRang... | Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro.h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
i : ι
a✝ : i ∈ a.support
⊢ ↑(if h : a i ∈ I then { val := a i, property := h } else 0) • v i = a 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 [dif_pos (ha i)] | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
refine' ⟨a.mapRang... | Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
case h.intro.intro
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
hv : Submodule.span R (Set.range v) = ⊤
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ ∀ (a : ι), ↑0 • v a = 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... | exact fun _ => zero_smul _ _ | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine' ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, _⟩
rintro ⟨a, ha, rfl⟩
classical
refine' ⟨a.mapRang... | Mathlib.RingTheory.Ideal.Operations.1961_0.5qK551sG47yBciY | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommSemiring R
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
b : Basis ι R S
x : S
hx : x ≠ 0
⊢ LinearMap.range ((Algebra.lmul R S) x) = Submodule.restrictScalars R (span {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... | ext | /-- A basis on `S` gives a basis on `Ideal.span {x}`, by multiplying everything by `x`. -/
noncomputable def basisSpanSingleton (b : Basis ι R S) {x : S} (hx : x ≠ 0) :
Basis ι R (span ({x} : Set S)) :=
b.map <|
LinearEquiv.ofInjective (Algebra.lmul R S x) (LinearMap.mul_injective hx) ≪≫ₗ
LinearEquiv.... | Mathlib.RingTheory.Ideal.Operations.1982_0.5qK551sG47yBciY | /-- A basis on `S` gives a basis on `Ideal.span {x}`, by multiplying everything by `x`. -/
noncomputable def basisSpanSingleton (b : Basis ι R S) {x : S} (hx : x ≠ 0) :
Basis ι R (span ({x} : Set S)) | Mathlib_RingTheory_Ideal_Operations |
case h
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommSemiring R
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
b : Basis ι R S
x : S
hx : x ≠ 0
x✝ : S
⊢ x✝ ∈ LinearMap.range ((Algebra.lmul R S) x) ↔ x✝ ∈ Submodule.restrictScalars R (span {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... | simp [mem_span_singleton', mul_comm] | /-- A basis on `S` gives a basis on `Ideal.span {x}`, by multiplying everything by `x`. -/
noncomputable def basisSpanSingleton (b : Basis ι R S) {x : S} (hx : x ≠ 0) :
Basis ι R (span ({x} : Set S)) :=
b.map <|
LinearEquiv.ofInjective (Algebra.lmul R S x) (LinearMap.mul_injective hx) ≪≫ₗ
LinearEquiv.... | Mathlib.RingTheory.Ideal.Operations.1982_0.5qK551sG47yBciY | /-- A basis on `S` gives a basis on `Ideal.span {x}`, by multiplying everything by `x`. -/
noncomputable def basisSpanSingleton (b : Basis ι R S) {x : S} (hx : x ≠ 0) :
Basis ι R (span ({x} : Set S)) | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommSemiring R
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
b : Basis ι R S
x : S
hx : x ≠ 0
i : ι
⊢ ↑((basisSpanSingleton b hx) i) = x * 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... | simp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply,
Submodule.restrictScalarsEquiv_apply, LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply,
LinearEquiv.restrictScalars_apply, Algebra.coe_lmul_eq_mul, LinearMap.mul_apply'] | @[simp]
theorem basisSpanSingleton_apply (b : Basis ι R S) {x : S} (hx : x ≠ 0) (i : ι) :
(basisSpanSingleton b hx i : S) = x * b i := by
| Mathlib.RingTheory.Ideal.Operations.1994_0.5qK551sG47yBciY | @[simp]
theorem basisSpanSingleton_apply (b : Basis ι R S) {x : S} (hx : x ≠ 0) (i : ι) :
(basisSpanSingleton b hx i : S) = x * b i | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommSemiring R
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
b : Basis ι R S
x : S
hx : x ≠ 0
i : ι
⊢ ↑((LinearEquiv.ofEq (LinearMap.range ((Algebra.lmul R S) x)) (Submodule.restrictScalars R (span {x}))
(_ : LinearMap.range ((Algebra.lmul R S) 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... | erw [LinearEquiv.coe_ofEq_apply, LinearEquiv.ofInjective_apply, Algebra.coe_lmul_eq_mul,
LinearMap.mul_apply'] | @[simp]
theorem basisSpanSingleton_apply (b : Basis ι R S) {x : S} (hx : x ≠ 0) (i : ι) :
(basisSpanSingleton b hx i : S) = x * b i := by
simp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply,
Submodule.restrictScalarsEquiv_apply, LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply,
... | Mathlib.RingTheory.Ideal.Operations.1994_0.5qK551sG47yBciY | @[simp]
theorem basisSpanSingleton_apply (b : Basis ι R S) {x : S} (hx : x ≠ 0) (i : ι) :
(basisSpanSingleton b hx i : S) = x * b i | Mathlib_RingTheory_Ideal_Operations |
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
N : Type u_4
inst✝² : Semiring N
inst✝¹ : Module N S
inst✝ : SMulCommClass R N S
b : Basis ι R S
x : S
hx : x ≠ 0
i : ι
⊢ (AddHom.toFun (↑(Basis.constr b N)).toAddHom (Subtype.val ∘ ⇑(basisSpanSin... | /-
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... | erw [Basis.constr_basis, Function.comp_apply, basisSpanSingleton_apply, LinearMap.mul_apply'] | @[simp]
theorem constr_basisSpanSingleton {N : Type*} [Semiring N] [Module N S] [SMulCommClass R N S]
(b : Basis ι R S) {x : S} (hx : x ≠ 0) :
(b.constr N).toFun (((↑) : _ → S) ∘ (basisSpanSingleton b hx)) = Algebra.lmul R S x :=
b.ext fun i => by
| Mathlib.RingTheory.Ideal.Operations.2005_0.5qK551sG47yBciY | @[simp]
theorem constr_basisSpanSingleton {N : Type*} [Semiring N] [Module N S] [SMulCommClass R N S]
(b : Basis ι R S) {x : S} (hx : x ≠ 0) :
(b.constr N).toFun (((↑) : _ → S) ∘ (basisSpanSingleton b hx)) = Algebra.lmul R S x | Mathlib_RingTheory_Ideal_Operations |
R : Type u_1
inst✝ : CommSemiring R
r : R
⊢ Associates.mk (Ideal.span {r}) ≠ 0 ↔ r ≠ 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... | rw [Associates.mk_ne_zero, Ideal.zero_eq_bot, Ne.def, Ideal.span_singleton_eq_bot] | theorem Associates.mk_ne_zero' {R : Type*} [CommSemiring R] {r : R} :
Associates.mk (Ideal.span {r} : Ideal R) ≠ 0 ↔ r ≠ 0 := by
| Mathlib.RingTheory.Ideal.Operations.2032_0.5qK551sG47yBciY | theorem Associates.mk_ne_zero' {R : Type*} [CommSemiring R] {r : R} :
Associates.mk (Ideal.span {r} : Ideal R) ≠ 0 ↔ r ≠ 0 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
G : Type u_2
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Semiring T
rcf : RingHomClass F R S
rcg : RingHomClass G T S
f : F
g : G
r : R
⊢ r ∈ ker f ↔ f r = 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... | rw [ker, Ideal.mem_comap, Submodule.mem_bot] | /-- An element is in the kernel if and only if it maps to zero.-/
theorem mem_ker {r} : r ∈ ker f ↔ f r = 0 := by | Mathlib.RingTheory.Ideal.Operations.2067_0.5qK551sG47yBciY | /-- An element is in the kernel if and only if it maps to zero.-/
theorem mem_ker {r} : r ∈ ker f ↔ f r = 0 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
G : Type u_2
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Semiring T
rcf : RingHomClass F R S
rcg : RingHomClass G T S
f✝ : F
g✝ : G
f : S →+* R
g : T →+* S
⊢ Ideal.comap g (ker f) = ker (comp f g) | /-
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_comap_bot, Ideal.comap_comap, RingHom.ker_eq_comap_bot] | theorem comap_ker (f : S →+* R) (g : T →+* S) : f.ker.comap g = ker (f.comp g) := by
| Mathlib.RingTheory.Ideal.Operations.2079_0.5qK551sG47yBciY | theorem comap_ker (f : S →+* R) (g : T →+* S) : f.ker.comap g = ker (f.comp g) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
G : Type u_2
inst✝³ : Semiring R
inst✝² : Semiring S
inst✝¹ : Semiring T
rcf : RingHomClass F R S
rcg : RingHomClass G T S
f✝ : F
g : G
inst✝ : Nontrivial 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... | rw [mem_ker, map_one] | /-- If the target is not the zero ring, then one is not in the kernel.-/
theorem not_one_mem_ker [Nontrivial S] (f : F) : (1 : R) ∉ ker f := by
| Mathlib.RingTheory.Ideal.Operations.2083_0.5qK551sG47yBciY | /-- If the target is not the zero ring, then one is not in the kernel.-/
theorem not_one_mem_ker [Nontrivial S] (f : F) : (1 : R) ∉ ker f | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
G : Type u_2
inst✝³ : Semiring R
inst✝² : Semiring S
inst✝¹ : Semiring T
rcf : RingHomClass F R S
rcg : RingHomClass G T S
f✝ : F
g : G
inst✝ : Nontrivial S
f : F
⊢ ¬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... | exact one_ne_zero | /-- If the target is not the zero ring, then one is not in the kernel.-/
theorem not_one_mem_ker [Nontrivial S] (f : F) : (1 : R) ∉ ker f := by
rw [mem_ker, map_one]
| Mathlib.RingTheory.Ideal.Operations.2083_0.5qK551sG47yBciY | /-- If the target is not the zero ring, then one is not in the kernel.-/
theorem not_one_mem_ker [Nontrivial S] (f : F) : (1 : R) ∉ ker f | Mathlib_RingTheory_Ideal_Operations |
R✝ : Type u
S : Type v
T : Type w
F : Type u_1
G : Type u_2
inst✝³ : Semiring R✝
inst✝² : Semiring S
inst✝¹ : Semiring T
rcf : RingHomClass F R✝ S
rcg : RingHomClass G T S
f : F
g : G
ι : Type u_3
R : ι → Type u_4
inst✝ : (i : ι) → Semiring (R i)
φ : (i : ι) → S →+* R i
⊢ ker (Pi.ringHom φ) = ⨅ i, ker (φ 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... | ext x | lemma _root_.Pi.ker_ringHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
(φ : ∀ i, S →+* R i) : ker (Pi.ringHom φ) = ⨅ i, ker (φ i) := by
| Mathlib.RingTheory.Ideal.Operations.2093_0.5qK551sG47yBciY | lemma _root_.Pi.ker_ringHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
(φ : ∀ i, S →+* R i) : ker (Pi.ringHom φ) = ⨅ i, ker (φ i) | Mathlib_RingTheory_Ideal_Operations |
case h
R✝ : Type u
S : Type v
T : Type w
F : Type u_1
G : Type u_2
inst✝³ : Semiring R✝
inst✝² : Semiring S
inst✝¹ : Semiring T
rcf : RingHomClass F R✝ S
rcg : RingHomClass G T S
f : F
g : G
ι : Type u_3
R : ι → Type u_4
inst✝ : (i : ι) → Semiring (R i)
φ : (i : ι) → S →+* R i
x : S
⊢ x ∈ ker (Pi.ringHom φ) ↔ x ∈ ⨅ 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... | simp [mem_ker, Ideal.mem_iInf, Function.funext_iff] | lemma _root_.Pi.ker_ringHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
(φ : ∀ i, S →+* R i) : ker (Pi.ringHom φ) = ⨅ i, ker (φ i) := by
ext x
| Mathlib.RingTheory.Ideal.Operations.2093_0.5qK551sG47yBciY | lemma _root_.Pi.ker_ringHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
(φ : ∀ i, S →+* R i) : ker (Pi.ringHom φ) = ⨅ i, ker (φ i) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝¹ : Ring R
inst✝ : Semiring S
rc : RingHomClass F R S
f : F
⊢ Function.Injective ⇑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 [SetLike.ext'_iff, ker_eq, Set.ext_iff] | theorem injective_iff_ker_eq_bot : Function.Injective f ↔ ker f = ⊥ := by
| Mathlib.RingTheory.Ideal.Operations.2104_0.5qK551sG47yBciY | theorem injective_iff_ker_eq_bot : Function.Injective f ↔ ker f = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝¹ : Ring R
inst✝ : Semiring S
rc : RingHomClass F R S
f : F
⊢ Function.Injective ⇑f ↔ ∀ (x : R), x ∈ ⇑f ⁻¹' {0} ↔ 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... | exact injective_iff_map_eq_zero' f | theorem injective_iff_ker_eq_bot : Function.Injective f ↔ ker f = ⊥ := by
rw [SetLike.ext'_iff, ker_eq, Set.ext_iff]
| Mathlib.RingTheory.Ideal.Operations.2104_0.5qK551sG47yBciY | theorem injective_iff_ker_eq_bot : Function.Injective f ↔ ker f = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝¹ : Ring R
inst✝ : Semiring S
rc : RingHomClass F R S
f : F
⊢ ker f = ⊥ ↔ ∀ (x : R), f x = 0 → 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... | rw [← injective_iff_map_eq_zero f, injective_iff_ker_eq_bot] | theorem ker_eq_bot_iff_eq_zero : ker f = ⊥ ↔ ∀ x, f x = 0 → x = 0 := by
| Mathlib.RingTheory.Ideal.Operations.2109_0.5qK551sG47yBciY | theorem ker_eq_bot_iff_eq_zero : ker f = ⊥ ↔ ∀ x, f x = 0 → x = 0 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝¹ : Ring R
inst✝ : Semiring S
rc : RingHomClass F R S
f✝ : F
f : R ≃+* S
⊢ 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 [← injective_iff_ker_eq_bot] using EquivLike.injective f | @[simp]
theorem ker_coe_equiv (f : R ≃+* S) : ker (f : R →+* S) = ⊥ := by
| Mathlib.RingTheory.Ideal.Operations.2113_0.5qK551sG47yBciY | @[simp]
theorem ker_coe_equiv (f : R ≃+* S) : ker (f : R →+* S) = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝² : Ring R
inst✝¹ : Semiring S
rc : RingHomClass F R S
f✝ : F
F' : Type u_2
inst✝ : RingEquivClass 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... | simpa only [← injective_iff_ker_eq_bot] using EquivLike.injective f | @[simp]
theorem ker_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') : ker f = ⊥ := by
| Mathlib.RingTheory.Ideal.Operations.2118_0.5qK551sG47yBciY | @[simp]
theorem ker_equiv {F' : Type*} [RingEquivClass F' R S] (f : F') : ker f = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
S : Type v
T : Type w
F : Type u_1
inst✝¹ : Ring R
inst✝ : Ring S
rc : RingHomClass F R S
f : F
x y : R
⊢ x - y ∈ ker f ↔ f x = f y | /-
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, map_sub, sub_eq_zero] | theorem sub_mem_ker_iff {x y} : x - y ∈ ker f ↔ f x = f y := by | Mathlib.RingTheory.Ideal.Operations.2129_0.5qK551sG47yBciY | theorem sub_mem_ker_iff {x y} : x - y ∈ ker f ↔ f x = f y | Mathlib_RingTheory_Ideal_Operations |
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