chore(algebra/module): Move submodule to its own file (#3696)

Author: eric-wieser

src/algebra/module/basic.lean

@@ -25,18 +25,13 @@ In this file we define
 
 * `is_linear_map R f` : predicate saying that `f : M → M₂` is a linear map.
 
-* `submodule R M` : a subset of `M` that contains zero and is closed with respect to addition and
-  scalar multiplication.
-
-* `subspace k M` : an abbreviation for `submodule` assuming that `k` is a `field`.
-
 ## Implementation notes
 
 * `vector_space` and `module` are abbreviations for `semimodule R M`.
 
 ## Tags
 
-semimodule, module, vector space, submodule, subspace, linear map
+semimodule, module, vector space, linear map
 -/
 
 open function
@@ -417,157 +412,6 @@ end is_linear_map
 abbreviation module.End (R : Type u) (M : Type v)
   [semiring R] [add_comm_monoid M] [semimodule R M] := M →ₗ[R] M
 
-set_option old_structure_cmd true
-
-/-- A submodule of a module is one which is closed under vector operations.
-  This is a sufficient condition for the subset of vectors in the submodule
-  to themselves form a module. -/
-structure submodule (R : Type u) (M : Type v) [semiring R]
-  [add_comm_monoid M] [semimodule R M] extends add_submonoid M : Type v :=
-(smul_mem' : ∀ (c:R) {x}, x ∈ carrier → c • x ∈ carrier)
-
-/-- Reinterpret a `submodule` as an `add_submonoid`. -/
-add_decl_doc submodule.to_add_submonoid
-
-namespace submodule
-
-variables [semiring R] [add_comm_monoid M] [semimodule R M]
-
-instance : has_coe_t (submodule R M) (set M) := ⟨λ s, s.carrier⟩
-instance : has_mem M (submodule R M) := ⟨λ x p, x ∈ (p : set M)⟩
-instance : has_coe_to_sort (submodule R M) := ⟨_, λ p, {x : M // x ∈ p}⟩
-
-variables (p q : submodule R M)
-
-@[simp, norm_cast] theorem coe_sort_coe : ↥(p : set M) = p := rfl
-
-variables {p q}
-
-protected theorem «exists» {q : p → Prop} : (∃ x, q x) ↔ (∃ x ∈ p, q ⟨x, ‹_›⟩) := set_coe.exists
-
-protected theorem «forall» {q : p → Prop} : (∀ x, q x) ↔ (∀ x ∈ p, q ⟨x, ‹_›⟩) := set_coe.forall
-
-theorem coe_injective : injective (coe : submodule R M → set M) :=
-λ p q h, by cases p; cases q; congr'
-
-@[simp, norm_cast] theorem coe_set_eq : (p : set M) = q ↔ p = q := coe_injective.eq_iff
-
-theorem ext'_iff : p = q ↔ (p : set M) = q := coe_set_eq.symm
-
-@[ext] theorem ext (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q := coe_injective $ set.ext h
-
-theorem to_add_submonoid_injective :
-  injective (to_add_submonoid : submodule R M → add_submonoid M) :=
-λ p q h, ext'_iff.2 $ add_submonoid.ext'_iff.1 h
-
-@[simp] theorem to_add_submonoid_eq : p.to_add_submonoid = q.to_add_submonoid ↔ p = q :=
-to_add_submonoid_injective.eq_iff
-
-end submodule
-
-namespace submodule
-
-section add_comm_monoid
-
-variables [semiring R] [add_comm_monoid M]
-
--- We can infer the module structure implicitly from the bundled submodule,
--- rather than via typeclass resolution.
-variables {semimodule_M : semimodule R M}
-variables {p q : submodule R M}
-variables {r : R} {x y : M}
-
-variables (p)
-@[simp] theorem mem_coe : x ∈ (p : set M) ↔ x ∈ p := iff.rfl
-
-@[simp] lemma zero_mem : (0 : M) ∈ p := p.zero_mem'
-
-lemma add_mem (h₁ : x ∈ p) (h₂ : y ∈ p) : x + y ∈ p := p.add_mem' h₁ h₂
-
-lemma smul_mem (r : R) (h : x ∈ p) : r • x ∈ p := p.smul_mem' r h
-
-lemma sum_mem {t : finset ι} {f : ι → M} : (∀c∈t, f c ∈ p) → (∑ i in t, f i) ∈ p :=
-p.to_add_submonoid.sum_mem
-
-lemma sum_smul_mem {t : finset ι} {f : ι → M} (r : ι → R)
-    (hyp : ∀ c ∈ t, f c ∈ p) : (∑ i in t, r i • f i) ∈ p :=
-submodule.sum_mem _ (λ i hi, submodule.smul_mem  _ _ (hyp i hi))
-
-@[simp] lemma smul_mem_iff' (u : units R) : (u:R) • x ∈ p ↔ x ∈ p :=
-⟨λ h, by simpa only [smul_smul, u.inv_mul, one_smul] using p.smul_mem ↑u⁻¹ h, p.smul_mem u⟩
-
-instance : has_add p := ⟨λx y, ⟨x.1 + y.1, add_mem _ x.2 y.2⟩⟩
-instance : has_zero p := ⟨⟨0, zero_mem _⟩⟩
-instance : inhabited p := ⟨0⟩
-instance : has_scalar R p := ⟨λ c x, ⟨c • x.1, smul_mem _ c x.2⟩⟩
-
-@[simp] lemma mk_eq_zero {x} (h : x ∈ p) : (⟨x, h⟩ : p) = 0 ↔ x = 0 := subtype.ext_iff_val
-
-variables {p}
-@[simp, norm_cast] lemma coe_eq_coe {x y : p} : (x : M) = y ↔ x = y := subtype.ext_iff_val.symm
-@[simp, norm_cast] lemma coe_eq_zero {x : p} : (x : M) = 0 ↔ x = 0 := @coe_eq_coe _ _ _ _ _ _ x 0
-@[simp, norm_cast] lemma coe_add (x y : p) : (↑(x + y) : M) = ↑x + ↑y := rfl
-@[simp, norm_cast] lemma coe_zero : ((0 : p) : M) = 0 := rfl
-@[simp, norm_cast] lemma coe_smul (r : R) (x : p) : ((r • x : p) : M) = r • ↑x := rfl
-@[simp, norm_cast] lemma coe_mk (x : M) (hx : x ∈ p) : ((⟨x, hx⟩ : p) : M) = x := rfl
-@[simp] lemma coe_mem (x : p) : (x : M) ∈ p := x.2
-
-@[simp] protected lemma eta (x : p) (hx : (x : M) ∈ p) : (⟨x, hx⟩ : p) = x := subtype.eta x hx
-
-variables (p)
-
-instance : add_comm_monoid p :=
-{ add := (+), zero := 0, .. p.to_add_submonoid.to_add_comm_monoid }
-
-instance : semimodule R p :=
-by refine {smul := (•), ..};
-   { intros, apply set_coe.ext, simp [smul_add, add_smul, mul_smul] }
-
-/-- Embedding of a submodule `p` to the ambient space `M`. -/
-protected def subtype : p →ₗ[R] M :=
-by refine {to_fun := coe, ..}; simp [coe_smul]
-
-@[simp] theorem subtype_apply (x : p) : p.subtype x = x := rfl
-
-lemma subtype_eq_val : ((submodule.subtype p) : p → M) = subtype.val := rfl
-
-end add_comm_monoid
-
-section add_comm_group
-
-variables [ring R] [add_comm_group M]
-variables {semimodule_M : semimodule R M}
-variables (p p' : submodule R M)
-variables {r : R} {x y : M}
-
-lemma neg_mem (hx : x ∈ p) : -x ∈ p := by rw ← neg_one_smul R; exact p.smul_mem _ hx
-
-/-- Reinterpret a submodule as an additive subgroup. -/
-def to_add_subgroup : add_subgroup M :=
-{ neg_mem' := λ _, p.neg_mem , .. p.to_add_submonoid }
-
-@[simp] lemma coe_to_add_subgroup : (p.to_add_subgroup : set M) = p := rfl
-
-lemma sub_mem : x ∈ p → y ∈ p → x - y ∈ p := p.to_add_subgroup.sub_mem
-
-@[simp] lemma neg_mem_iff : -x ∈ p ↔ x ∈ p := p.to_add_subgroup.neg_mem_iff
-
-lemma add_mem_iff_left : y ∈ p → (x + y ∈ p ↔ x ∈ p) := p.to_add_subgroup.add_mem_cancel_right
-
-lemma add_mem_iff_right : x ∈ p → (x + y ∈ p ↔ y ∈ p) := p.to_add_subgroup.add_mem_cancel_left
-
-instance : has_neg p := ⟨λx, ⟨-x.1, neg_mem _ x.2⟩⟩
-
-@[simp, norm_cast] lemma coe_neg (x : p) : ((-x : p) : M) = -x := rfl
-
-instance : add_comm_group p :=
-{ add := (+), zero := 0, neg := has_neg.neg, ..p.to_add_subgroup.to_add_comm_group }
-
-@[simp, norm_cast] lemma coe_sub (x y : p) : (↑(x - y) : M) = ↑x - ↑y := rfl
-
-end add_comm_group
-
-end submodule
 
 /--
 Vector spaces are defined as an `abbreviation` for semimodules,
@@ -591,21 +435,6 @@ library_note "vector space definition"
 abbreviation vector_space (R : Type u) (M : Type v) [field R] [add_comm_group M] :=
 semimodule R M
 
-/-- Subspace of a vector space. Defined to equal `submodule`. -/
-abbreviation subspace (R : Type u) (M : Type v)
-  [field R] [add_comm_group M] [vector_space R M] :=
-submodule R M
-
-namespace submodule
-
-variables [division_ring R] [add_comm_group M] [module R M]
-variables (p : submodule R M) {r : R} {x y : M}
-
-theorem smul_mem_iff (r0 : r ≠ 0) : r • x ∈ p ↔ x ∈ p :=
-p.smul_mem_iff' (units.mk0 r r0)
-
-end submodule
-
 namespace add_comm_monoid
 open add_monoid
 

src/algebra/module/default.lean

@@ -4,8 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
 import algebra.module.basic
+import algebra.module.submodule
 
 /-!
 # Default file for module
-This file imports `algebra.module.basic`
+This file imports `algebra.module.basic` and `algebra.module.submodule`.
 -/