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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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 rw [Submodule.mem_annihilator]
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 rw [Submodule.mem_annihilator] constructor ·
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 rw [Submodule.mem_annihilator] constructor · intro h n
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) ·
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ ·
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 exact h ⟨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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ ·
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 exact h ⟨x, hx⟩ · exact smul_zero _ ·
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy
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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] ·
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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] ·
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx
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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a 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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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
@[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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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
@[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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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
@[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 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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨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 ∈ ↑N hj' : ((LinearMap.lsmul R M) ↑{ val := i, property := hi }) j = m ⊢ p 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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩
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 ⟨j, hj, hj'⟩
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 ∈ ↑N hj' : ((LinearMap.lsmul R M) ↑{ val := i, property := hi }) j = m ⊢ p (((LinearMap.lsmul R M) ↑{ val := i, property := hi }) 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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj']
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 ⟨j, hj, hj'⟩ rw [← hj']
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) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) (_ : x + y ∈ I • N) ⊢ p x 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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by
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 _ ‹_› ‹_›)) : p x hx := by
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 _ ‹_› ‹_›)) : p x hx
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) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) (_ : x + y ∈ I • N) ⊢ ∃ (x_1 : x ∈ I • N), p x x_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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H
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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H
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 _ ‹_› ‹_›)) : p x hx
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 : y2 • m = m2 ⊢ (y1 + y2) • m = m1 + m2
/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by
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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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⟩
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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]
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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]
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.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
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.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)
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.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
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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.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 => 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.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' 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 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 exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[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 ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- 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 _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' 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, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right 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] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.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 =>
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

miniCTX: Neural Theorem Proving with (Long-)Contexts

Lean 4 tactic prediction examples extracted from Mathlib.

These examples have not been formatted for instruction tuning (including data splits).

Please see l3lab/ntp-mathlib-instruct-* for datasets with instruction tuning examples.

Version

Generated using ntptoolkit's ntp-training-data.

It used the following config for ntp-training-data:

{
    "repo": "https://github.com/leanprover-community/mathlib4",
    "commit": "cf8e23a62939ed7cc530fbb68e83539730f32f86",
    "lean": "leanprover/lean4:v4.4.0",
    "name": "mathlib",
    "import_file": "Mathlib.lean",
    "imports": ["Mathlib"]
}

Example usage:

ds = datasets.load_dataset('l3lab/ntp-mathlib')

print(len(ds['train']))
# ==> 307049

Format:

{
  'state': 'proof state',
  'srcUpToTactic': 'source up to tactic invocation',
  'nextTactic': 'tactic',
  'declUpToTactic': 'declariation up to tactic invocation',
  'declId': 'unique ID for declaration',
  'decl': 'declaration',
  'file_tag': 'file ID'
}

Citation

Please cite:

@misc{hu2024minictx,
  author = {Jiewen Hu and Thomas Zhu and Sean Welleck},
  title = {miniCTX: Neural Theorem Proving with (Long-)Contexts},
  year = {2024},
  eprint={},
  archivePrefix={arXiv},
}
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