state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
⊢ r ∈ annihilator (span R s) ↔ ∀ (n : ↑s), r • ↑n = 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 [Submodule.mem_annihilator] | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
| Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
⊢ (∀ n ∈ span R s, r • n = 0) ↔ ∀ (n : ↑s), r • ↑n = 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... | constructor | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
| Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
⊢ (∀ n ∈ span R s, r • n = 0) → ∀ (n : ↑s), r • ↑n = 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... | intro h n | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ n ∈ span R s, r • n = 0
n : ↑s
⊢ r • ↑n = 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 h _ (Submodule.subset_span n.prop) | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
| Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
⊢ (∀ (n : ↑s), r • ↑n = 0) → ∀ n ∈ span R s, r • n = 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... | intro h n hn | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
⊢ r • n = 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... | refine Submodule.span_induction hn ?_ ?_ ?_ ?_ | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
| Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
⊢ ∀ x ∈ s, r • 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... | intro x hx | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
x : M
hx : x ∈ s
⊢ r • 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... | exact h ⟨x, hx⟩ | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
... | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
⊢ r • 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... | exact smul_zero _ | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
... | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_3
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
⊢ ∀ (x y : M), r • x = 0 → r • y = 0 → r • (x + y) = 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... | intro x y hx hy | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
... | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_3
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
x y : M
hx : r • x = 0
hy : r • y = 0
⊢ r • (x + y) = 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 [smul_add, hx, hy, zero_add] | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
... | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
⊢ ∀ (a : R) (x : M), r • x = 0 → r • a • 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... | intro a x hx | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
... | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
case mpr.refine_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
s : Set M
r : R
h : ∀ (n : ↑s), r • ↑n = 0
n : M
hn : n ∈ span R s
a : R
x : M
hx : r • x = 0
⊢ r • a • 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 [smul_comm, hx, smul_zero] | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
... | Mathlib.RingTheory.Ideal.Operations.60_0.5qK551sG47yBciY | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
g : M
r : R
⊢ r ∈ annihilator (span R {g}) ↔ r • g = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simp [mem_annihilator_span] | theorem mem_annihilator_span_singleton (g : M) (r : R) :
r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by | Mathlib.RingTheory.Ideal.Operations.77_0.5qK551sG47yBciY | theorem mem_annihilator_span_singleton (g : M) (r : R) :
r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
p : M → Prop
x : M
H : x ∈ I • N
Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)
H1 : ∀ (x y : M), p x → p y → p (x + y)
⊢ p 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... | have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x := by
| Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
p : M → Prop
x : M
H : x ∈ I • N
Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)
H1 : ∀ (x y : M), p x → p y → p (x + y)
⊢ p 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... | simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x := by
have H0 : p 0 := by | Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
p : M → Prop
x : M
H : x ∈ I • N
Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)
H1 : ∀ (x y : M), p x → p y → p (x + y)
H0 : p 0
⊢ p 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.iSup_induction (x := x) _ H ?_ H0 H1 | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x := by
have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem
| Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
p : M → Prop
x : M
H : x ∈ I • N
Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)
H1 : ∀ (x y : M), p x → p y → p (x + y)
H0 : p 0
⊢ ∀ (i : ↥I), ∀ x ∈ map ((LinearMap.lsmul R M) ↑i) N, p... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x := by
have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem
refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1
| Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | Mathlib_RingTheory_Ideal_Operations |
case mk.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
p : M → Prop
x : M
H : x ∈ I • N
Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)
H1 : ∀ (x y : M), p x → p y → p (x + y)
H0 : p 0
i : R
hi : i ∈ I
m j : M
hj : 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 [← hj'] | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x := by
have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem
refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1
rintro ⟨i, hi⟩ m ⟨... | Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | Mathlib_RingTheory_Ideal_Operations |
case mk.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
p : M → Prop
x : M
H : x ∈ I • N
Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)
H1 : ∀ (x y : M), p x → p y → p (x + y)
H0 : p 0
i : R
hi : i ∈ I
m j : M
hj : 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 Hb _ hi _ hj | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x := by
have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem
refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1
rintro ⟨i, hi⟩ m ⟨... | Mathlib.RingTheory.Ideal.Operations.113_0.5qK551sG47yBciY | @[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
x : M
hx : x ∈ I • N
p : (x : M) → x ∈ I • N → Prop
Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N)
H1 : ∀ (x : M) (hx : x ∈ I • N) (y : M... | /-
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' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H | /-- Dependent version of `Submodule.smul_induction_on`. -/
@[elab_as_elim]
theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹... | Mathlib.RingTheory.Ideal.Operations.123_0.5qK551sG47yBciY | /-- Dependent version of `Submodule.smul_induction_on`. -/
@[elab_as_elim]
theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹... | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
x : M
hx : x ∈ I • N
p : (x : M) → x ∈ I • N → Prop
Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N)
H1 : ∀ (x : M) (hx : x ∈ I • N) (y : M... | /-
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
smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ =>
⟨_, H1 _ _ _ _ hx hy⟩ | /-- Dependent version of `Submodule.smul_induction_on`. -/
@[elab_as_elim]
theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹... | Mathlib.RingTheory.Ideal.Operations.123_0.5qK551sG47yBciY | /-- Dependent version of `Submodule.smul_induction_on`. -/
@[elab_as_elim]
theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹... | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I✝ J : Ideal R
N P : Submodule R M
I : Ideal R
m x : M
hx : x ∈ I • span R {m}
m1 m2 : M
x✝¹ : ∃ y ∈ I, y • m = m1
x✝ : ∃ y ∈ I, y • m = m2
y1 : R
hyi1 : y1 ∈ I
hy1 : y1 • m = m1
y2 : R
hyi2 : y2 ∈ I
hy2 ... | /-
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 [add_smul, hy1, hy2] | theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} :
x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x :=
⟨fun hx =>
smul_induction_on hx
(fun r hri n hnm =>
let ⟨s, hs⟩ := mem_span_singleton.1 hnm
⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩)
fun m1 m2 ⟨y1, hyi1,... | Mathlib.RingTheory.Ideal.Operations.134_0.5qK551sG47yBciY | theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} :
x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I✝ J : Ideal R
N P : Submodule R M
I : Ideal R
f : R →ₗ[R] M
⊢ 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... | rintro _ ⟨y, hy, rfl⟩ | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) := by
| Mathlib.RingTheory.Ideal.Operations.162_0.5qK551sG47yBciY | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I✝ J : Ideal R
N P : Submodule R M
I : Ideal R
f : R →ₗ[R] M
y : R
hy : y ∈ ↑I
⊢ f y ∈ 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 y, ← smul_eq_mul, f.map_smul] | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) := by
rintro _ ⟨y, hy, rfl⟩
| Mathlib.RingTheory.Ideal.Operations.162_0.5qK551sG47yBciY | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I✝ J : Ideal R
N P : Submodule R M
I : Ideal R
f : R →ₗ[R] M
y : R
hy : y ∈ ↑I
⊢ y • f 1 ∈ 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 smul_mem_smul hy mem_top | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) := by
rintro _ ⟨y, hy, rfl⟩
rw [← mul_one y, ← smul_eq_mul, f.map_smul]
| Mathlib.RingTheory.Ideal.Operations.162_0.5qK551sG47yBciY | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I✝ J : Ideal R
N P : Submodule R M
I : Ideal R
⊢ I * annihilator 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_comm, annihilator_mul] | @[simp]
theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by | Mathlib.RingTheory.Ideal.Operations.179_0.5qK551sG47yBciY | @[simp]
theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
⊢ Ideal.span {r} • N = r • 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... | have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
exact (Set.image_eq_iUnion _ (N : Set M)).symm | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
| Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
⊢ span R (⋃ t ∈ N, {r • t}) = r • 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... | convert span_eq (r • N) | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
| Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
case h.e'_2.h.e'_6
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
⊢ ⋃ t ∈ N, {r • t} = ↑(r • 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 (Set.image_eq_iUnion _ (N : Set M)).symm | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
| Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
this : span R (⋃ t ∈ N, {r • t}) = r • N
⊢ Ideal.span {r} • N = r • 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... | conv_lhs => rw [← span_eq N, span_smul_span] | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
exact (Set.image_eq_iUnion _ (N : Set M)).symm
| Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
this : span R (⋃ t ∈ N, {r • t}) = r • N
| Ideal.span {r} • 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 [← span_eq N, span_smul_span] | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
exact (Set.image_eq_iUnion _ (N : Set M)).symm
conv_lhs => | Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
this : span R (⋃ t ∈ N, {r • t}) = r • N
| Ideal.span {r} • 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 [← span_eq N, span_smul_span] | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
exact (Set.image_eq_iUnion _ (N : Set M)).symm
conv_lhs => | Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
this : span R (⋃ t ∈ N, {r • t}) = r • N
| Ideal.span {r} • 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 [← span_eq N, span_smul_span] | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
exact (Set.image_eq_iUnion _ (N : Set M)).symm
conv_lhs => | Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
r : R
N : Submodule R M
this : span R (⋃ t ∈ N, {r • t}) = r • N
⊢ span R (⋃ s ∈ {r}, ⋃ t ∈ ↑N, {s • t}) = r • 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... | simpa | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N := by
have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by
convert span_eq (r • N)
exact (Set.image_eq_iUnion _ (N : Set M)).symm
conv_lhs => rw [← span_eq N, span_smul_span]
| Mathlib.RingTheory.Ideal.Operations.240_0.5qK551sG47yBciY | theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
(Ideal.span {r} : Ideal R) • N = r • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ↑r • x ∈ M'
⊢ x ∈ M' | /-
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 (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by
rw [top_smul] at this
exact this (subset_span (Set.mem_singleton x)) | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by
| Mathlib.RingTheory.Ideal.Operations.249_0.5qK551sG47yBciY | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ↑r • x ∈ M'
this : ⊤ • span R {x} ≤ M'
⊢ x ∈ M' | /-
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 [top_smul] at this | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by
suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by
| Mathlib.RingTheory.Ideal.Operations.249_0.5qK551sG47yBciY | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ↑r • x ∈ M'
this : span R {x} ≤ M'
⊢ x ∈ M' | /-
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 this (subset_span (Set.mem_singleton x)) | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by
suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by
rw [top_smul] at this
| Mathlib.RingTheory.Ideal.Operations.249_0.5qK551sG47yBciY | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ↑r • x ∈ M'
⊢ ⊤ • span R {x} ≤ M' | /-
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 [← hs, span_smul_span, span_le] | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by
suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by
rw [top_smul] at this
exact this (subset_span (Set.mem_singleton x))
| Mathlib.RingTheory.Ideal.Operations.249_0.5qK551sG47yBciY | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ↑r • x ∈ M'
⊢ ⋃ s_1 ∈ s, ⋃ t ∈ {x}, {s_1 • t} ⊆ ↑M' | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | simpa using H | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by
suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by
rw [top_smul] at this
exact this (subset_span (Set.mem_singleton x))
rw [← hs, span_smul_span, span_le... | Mathlib.RingTheory.Ideal.Operations.249_0.5qK551sG47yBciY | theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ∃ n, ↑r ^ n • x ∈ M'
⊢ x ∈ M' | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
H : ∀ (r : ↑s), ∃ n, ↑r ^ n • x ∈ M'
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
H : ∀ (r : { x // x ∈ s... | /-
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... | choose n₁ n₂ using H | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ... | /-
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... | let N := s'.attach.sup n₁ | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } →... | /-
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 hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } →... | /-
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 M'.mem_of_span_top_of_smul_mem _ hs' | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro.H
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' }... | /-
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 ⟨_, r, hr, rfl⟩ | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case intro.intro.H.mk.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case h.e'_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂... | /-
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 [Subtype.coe_mk, smul_smul, ← pow_add] | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
case h.e'_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂... | /-
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 [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib.RingTheory.Ideal.Operations.258_0.5qK551sG47yBciY | /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ... | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
s : Set M
x : M
⊢ x ∈ I • span R s ↔ x ∈ span R (⋃ a ∈ I, ⋃ b ∈ s, {a • b}) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] | theorem mem_smul_span {s : Set M} {x : M} :
x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by
| Mathlib.RingTheory.Ideal.Operations.289_0.5qK551sG47yBciY | theorem mem_smul_span {s : Set M} {x : M} :
x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
s : Set M
x : M
⊢ x ∈ span R (⋃ s_1 ∈ ↑I, ⋃ t ∈ s, {s_1 • t}) ↔ x ∈ span R (⋃ a ∈ I, ⋃ b ∈ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rfl | theorem mem_smul_span {s : Set M} {x : M} :
x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by
rw [← I.span_eq, Submodule.span_smul_span, I.span_eq]
| Mathlib.RingTheory.Ideal.Operations.289_0.5qK551sG47yBciY | theorem mem_smul_span {s : Set M} {x : M} :
x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ I • span R (Set.range f) ↔ ∃ a, ∃ (_ : ∀ (i : ι), 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... | constructor | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ I • span R (Set.range f) → ∃ a, ∃ (_ : ∀ (i : ι)... | /-
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... | swap | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mpr
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fu... | /-
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⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mpr.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.s... | /-
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.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ I • span R (Set.range f) → ∃ a, ∃ (_ : ∀ (i : ι)... | /-
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 hx => span_induction (mem_smul_span.mp hx) _ _ _ _ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range 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... | simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ (x : M... | /-
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 ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1.intro.intro.intro.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • s... | /-
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' ⟨Finsupp.single i y, fun j => _, _⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx :... | /-
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... | letI := Classical.decEq ι | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx :... | /-
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.single_apply] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx :... | /-
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... | split_ifs | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case pos
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ 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... | assumption | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case neg
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ 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... | exact I.zero_mem | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx :... | /-
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' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx :... | /-
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 | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∃ a, ∃ (... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_3
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ (x 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... | rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_3.intro.intro.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span 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... | refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_3.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : 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... | intros | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_3.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : 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... | intros | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_3.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : 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 only [zero_smul, add_smul] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_3.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : 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 only [zero_smul, add_smul] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ (a : 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... | rintro c x ⟨a, ha, rfl⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range 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 [Finsupp.sum_smul_index, Finsupp.smul_sum] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range 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... | intros | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range 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... | intros | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range 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... | simp only [zero_smul, mul_smul] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range 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... | simp only [zero_smul, mul_smul] | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib.RingTheory.Ideal.Operations.297_0.5qK551sG47yBciY | /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (Set.range f) ↔
∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c =... | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ x ∈ I • span R (f '' s) ↔ ∃ a, ∃ (_ : ∀ (i : ↑s), ... | /-
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, ← Set.image_eq_range] | theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) :
x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x :=
by | Mathlib.RingTheory.Ideal.Operations.325_0.5qK551sG47yBciY | theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) :
x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
N : Submodule R M
x : ↥N
⊢ x ∈ I • ⊤ ↔ ↑x ∈ I • 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... | change _ ↔ N.subtype x ∈ I • N | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by
| Mathlib.RingTheory.Ideal.Operations.330_0.5qK551sG47yBciY | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
N : Submodule R M
x : ↥N
⊢ x ∈ I • ⊤ ↔ (Submodule.subtype N) x ∈ I • 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... | have : Submodule.map N.subtype (I • ⊤) = I • N := by
rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by
change _ ↔ N.subtype x ∈ I • N
| Mathlib.RingTheory.Ideal.Operations.330_0.5qK551sG47yBciY | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
N : Submodule R M
x : ↥N
⊢ map (Submodule.subtype N) (I • ⊤) = I • 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 [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by
change _ ↔ N.subtype x ∈ I • N
have : Submodule.map N.subtype (I • ⊤) = I • N := by
| Mathlib.RingTheory.Ideal.Operations.330_0.5qK551sG47yBciY | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
N : Submodule R M
x : ↥N
this : map (Submodule.subtype N) (I • ⊤) = I • N
⊢ x ∈ I • ⊤ ↔ (S... | /-
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 [← this] | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by
change _ ↔ N.subtype x ∈ I • N
have : Submodule.map N.subtype (I • ⊤) = I • N := by
rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]
| Mathlib.RingTheory.Ideal.Operations.330_0.5qK551sG47yBciY | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
N : Submodule R M
x : ↥N
this : map (Submodule.subtype N) (I • ⊤) = I • N
⊢ x ∈ I • ⊤ ↔ (S... | /-
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 (Function.Injective.mem_set_image N.injective_subtype).symm | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by
change _ ↔ N.subtype x ∈ I • N
have : Submodule.map N.subtype (I • ⊤) = I • N := by
rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]
rw [← this]
| Mathlib.RingTheory.Ideal.Operations.330_0.5qK551sG47yBciY | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I✝ J : Ideal R
N P : Submodule R M
S✝ : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
S : Submodule R M'
I : Ideal R
⊢ I • comap f S ≤ comap f (I • S) | /-
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_le.mpr fun r hr x hx => _ | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f := by
| Mathlib.RingTheory.Ideal.Operations.339_0.5qK551sG47yBciY | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I✝ J : Ideal R
N P : Submodule R M
S✝ : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
S : Submodule R M'
I : Ideal R
r : R
hr : r ∈ I
x : M
hx : x ∈ comap f S
⊢... | /-
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_comap] at hx ⊢ | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f := by
refine' Submodule.smul_le.mpr fun r hr x hx => _
| Mathlib.RingTheory.Ideal.Operations.339_0.5qK551sG47yBciY | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I✝ J : Ideal R
N P : Submodule R M
S✝ : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
S : Submodule R M'
I : Ideal R
r : R
hr : r ∈ I
x : M
hx : f x ∈ S
⊢ f (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 [f.map_smul] | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f := by
refine' Submodule.smul_le.mpr fun r hr x hx => _
rw [Submodule.mem_comap] at hx ⊢
| Mathlib.RingTheory.Ideal.Operations.339_0.5qK551sG47yBciY | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I✝ J : Ideal R
N P : Submodule R M
S✝ : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
S : Submodule R M'
I : Ideal R
r : R
hr : r ∈ I
x : M
hx : f x ∈ S
⊢ r • 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 Submodule.smul_mem_smul hr hx | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f := by
refine' Submodule.smul_le.mpr fun r hr x hx => _
rw [Submodule.mem_comap] at hx ⊢
rw [f.map_smul]
| Mathlib.RingTheory.Ideal.Operations.339_0.5qK551sG47yBciY | @[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
I • S.comap f ≤ (I • S).comap f | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N✝ N₁ N₂ P P₁ P₂ N : Submodule R M
x : M
r : R
⊢ r ∈ colon N (span R {x}) ↔ ∀ (a : R), r • a • x ∈ 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... | simp [Submodule.mem_colon, Submodule.mem_span_singleton] | @[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N :=
calc
r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by
| Mathlib.RingTheory.Ideal.Operations.389_0.5qK551sG47yBciY | @[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N✝ N₁ N₂ P P₁ P₂ N : Submodule R M
x : M
r : R
⊢ (∀ (a : R), r • a • x ∈ N) ↔ r • x ∈ 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... | simp_rw [fun (a : R) ↦ smul_comm r a x] | @[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N :=
calc
r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by
simp [Submodule.mem_colon, Submodule.mem_span_singleton]
_ ↔ r • x ∈ N := by | Mathlib.RingTheory.Ideal.Operations.389_0.5qK551sG47yBciY | @[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N✝ N₁ N₂ P P₁ P₂ N : Submodule R M
x : M
r : R
⊢ (∀ (a : R), a • r • x ∈ N) ↔ r • x ∈ 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 SetLike.forall_smul_mem_iff | @[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N :=
calc
r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by
simp [Submodule.mem_colon, Submodule.mem_span_singleton]
_ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_... | Mathlib.RingTheory.Ideal.Operations.389_0.5qK551sG47yBciY | @[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N | Mathlib_RingTheory_Ideal_Operations |
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N N₁ N₂ P P₁ P₂ : Submodule R M
I : Ideal R
x r : R
⊢ r ∈ colon I (Ideal.span {x}) ↔ r * 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 only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] | @[simp]
theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} :
r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by
| Mathlib.RingTheory.Ideal.Operations.398_0.5qK551sG47yBciY | @[simp]
theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} :
r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ 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... | erw [Submodule.one_eq_range, LinearMap.range_id] | @[simp]
theorem one_eq_top : (1 : Ideal R) = ⊤ := by | Mathlib.RingTheory.Ideal.Operations.440_0.5qK551sG47yBciY | @[simp]
theorem one_eq_top : (1 : Ideal R) = ⊤ | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Basis.Bilinear
import Mathlib.RingTh... | rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] | theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by
| Mathlib.RingTheory.Ideal.Operations.444_0.5qK551sG47yBciY | theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → Ideal R
x : ι → R
⊢ (∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, 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... | classical
refine Finset.induction_on s ?_ ?_
· intro
rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]
exact Submodule.mem_top
· intro a s ha IH h
rw [Finset.prod_insert ha, Finset.prod_insert ha]
exact
mul_mem_mul (h a <| Finset.mem_insert_self a s)
(IH fun i h... | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
| Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → Ideal R
x : ι → R
⊢ (∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, 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... | refine Finset.induction_on s ?_ ?_ | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
| Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
case refine_1
R : Type u
ι✝ : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
ι : Type u_2
s : Finset ι
I : ι → Ideal R
x : ι → R
⊢ (∀ i ∈ ∅, x i ∈ I i) → ∏ i in ∅, x i ∈ ∏ i in ∅, 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... | intro | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by
classical
refine Finset.induction_on s ?_ ?_
· | Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY | theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i | Mathlib_RingTheory_Ideal_Operations |
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