mathlib3 documentation

algebra.module.submodule.basic

Submodules of a module #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file we define

Tags #

submodule, subspace, linear map

source
def submodule.to_add_submonoid {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (self : submodule R M) :
add_submonoid M

Reinterpret a submodule as an add_submonoid.

source
structure submodule (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [module R M] :
Type v

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.

Instances for submodule
source
def submodule.to_sub_mul_action {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (self : submodule R M) :
sub_mul_action R M

Reinterpret a submodule as an sub_mul_action.

source
@[protected, instance]
def submodule.set_like {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
set_like (submodule R M) M
Equations
source
@[protected, instance]
def submodule.add_submonoid_class {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
add_submonoid_class (submodule R M) M
source
@[protected, instance]
def submodule.smul_mem_class {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
smul_mem_class (submodule R M) R M
Equations
source
@[simp]
theorem submodule.mem_to_add_submonoid {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) (x : M) :
x ∈ p.to_add_submonoid ↔ x ∈ p
source
@[simp]
theorem submodule.mem_mk {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {S : set M} {x : M} (h₁ : ∀ {a b : M}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : 0 ∈ S) (h₃ : ∀ (c : R) {x : M}, x ∈ S → c • x ∈ S) :
x ∈ {carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃} ↔ x ∈ S
source
@[simp]
theorem submodule.coe_set_mk {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (S : set M) (h₁ : ∀ {a b : M}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : 0 ∈ S) (h₃ : ∀ (c : R) {x : M}, x ∈ S → c • x ∈ S) :
↑{carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃} = S
source
@[simp]
theorem submodule.mk_le_mk {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {S S' : set M} (h₁ : ∀ {a b : M}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : 0 ∈ S) (h₃ : ∀ (c : R) {x : M}, x ∈ S → c • x ∈ S) (h₁' : ∀ {a b : M}, a ∈ S' → b ∈ S' → a + b ∈ S') (h₂' : 0 ∈ S') (h₃' : ∀ (c : R) {x : M}, x ∈ S' → c • x ∈ S') :
{carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃} ≤ {carrier := S', add_mem' := h₁', zero_mem' := h₂', smul_mem' := h₃'} ↔ S ⊆ S'
source
@[ext]
theorem submodule.ext {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {p q : submodule R M} (h : ∀ (x : M), x ∈ p ↔ x ∈ q) :
p = q
source
@[protected]
def submodule.copy {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) (s : set M) (hs : s = ↑p) :
submodule R M

Copy of a submodule with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations
source
@[simp]
theorem submodule.coe_copy {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (S : submodule R M) (s : set M) (hs : s = ↑S) :
↑(S.copy s hs) = s
source
theorem submodule.copy_eq {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (S : submodule R M) (s : set M) (hs : s = ↑S) :
S.copy s hs = S
source
theorem submodule.to_add_submonoid_injective {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
function.injective submodule.to_add_submonoid
source
@[simp]
theorem submodule.to_add_submonoid_eq {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {p q : submodule R M} :
p.to_add_submonoid = q.to_add_submonoid ↔ p = q
source
theorem submodule.to_add_submonoid_strict_mono {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
strict_mono submodule.to_add_submonoid
source
theorem submodule.to_add_submonoid_le {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {p q : submodule R M} :
p.to_add_submonoid ≤ q.to_add_submonoid ↔ p ≤ q
source
theorem submodule.to_add_submonoid_mono {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
monotone submodule.to_add_submonoid
source
@[simp]
theorem submodule.coe_to_add_submonoid {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
↑(p.to_add_submonoid) = ↑p
source
theorem submodule.to_sub_mul_action_injective {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
function.injective submodule.to_sub_mul_action
source
@[simp]
theorem submodule.to_sub_mul_action_eq {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {p q : submodule R M} :
p.to_sub_mul_action = q.to_sub_mul_action ↔ p = q
source
theorem submodule.to_sub_mul_action_strict_mono {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
strict_mono submodule.to_sub_mul_action
source
theorem submodule.to_sub_mul_action_mono {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] :
monotone submodule.to_sub_mul_action
source
@[simp]
theorem submodule.coe_to_sub_mul_action {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
↑(p.to_sub_mul_action) = ↑p
source
@[protected, instance]
def smul_mem_class.to_module {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {A : Type u_1} [set_like A M] [add_submonoid_class A M] [hA : smul_mem_class A R M] (S' : A) :
module R ↥S'

A submodule of a module is a module.

Equations
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@[protected]
def smul_mem_class.subtype {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {A : Type u_1} [set_like A M] [add_submonoid_class A M] [hA : smul_mem_class A R M] (S' : A) :
↥S' →ₗ[R] M

The natural R-linear map from a submodule of an R-module M to M.

Equations
source
@[protected, simp]
theorem smul_mem_class.coe_subtype {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] {A : Type u_1} [set_like A M] [add_submonoid_class A M] [hA : smul_mem_class A R M] (S' : A) :
⇑(smul_mem_class.subtype S') = coe
source
@[simp]
theorem submodule.mem_carrier {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {x : M} :
x ∈ p.carrier ↔ x ∈ ↑p
source
@[protected, simp]
theorem submodule.zero_mem {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
0 ∈ p
source
@[protected]
theorem submodule.add_mem {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {x y : M} (h₁ : x ∈ p) (h₂ : y ∈ p) :
x + y ∈ p
source
theorem submodule.smul_mem {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {x : M} (r : R) (h : x ∈ p) :
r • x ∈ p
source
theorem submodule.smul_of_tower_mem {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {x : M} [has_smul S R] [has_smul S M] [is_scalar_tower S R M] (r : S) (h : x ∈ p) :
r • x ∈ p
source
@[protected]
theorem submodule.sum_mem {R : Type u} {M : Type v} {ι : Type w} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {t : finset ι} {f : ι → M} :
(∀ (c : ι), c ∈ t → f c ∈ p) → t.sum (λ (i : ι), f i) ∈ p
source
theorem submodule.sum_smul_mem {R : Type u} {M : Type v} {ι : Type w} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {t : finset ι} {f : ι → M} (r : ι → R) (hyp : ∀ (c : ι), c ∈ t → f c ∈ p) :
t.sum (λ (i : ι), r i • f i) ∈ p
source
@[simp]
theorem submodule.smul_mem_iff' {G : Type u''} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {x : M} [group G] [mul_action G M] [has_smul G R] [is_scalar_tower G R M] (g : G) :
g • x ∈ p ↔ x ∈ p
source
@[protected, instance]
def submodule.has_add {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
has_add ↥p
Equations
source
@[protected, instance]
def submodule.has_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
has_zero ↥p
Equations
source
@[protected, instance]
def submodule.inhabited {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
inhabited ↥p
Equations
source
@[protected, instance]
def submodule.has_smul {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) [has_smul S R] [has_smul S M] [is_scalar_tower S R M] :
has_smul S ↥p
Equations
source
@[protected, instance]
def submodule.is_scalar_tower {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) [has_smul S R] [has_smul S M] [is_scalar_tower S R M] :
is_scalar_tower S R ↥p
source
@[protected, instance]
def submodule.is_scalar_tower' {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {S' : Type u_1} [has_smul S R] [has_smul S M] [has_smul S' R] [has_smul S' M] [has_smul S S'] [is_scalar_tower S' R M] [is_scalar_tower S S' M] [is_scalar_tower S R M] :
is_scalar_tower S S' ↥p
source
@[protected, instance]
def submodule.is_central_scalar {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) [has_smul S R] [has_smul S M] [is_scalar_tower S R M] [has_smul Sᵐᵒᵖ R] [has_smul Sᵐᵒᵖ M] [is_scalar_tower Sᵐᵒᵖ R M] [is_central_scalar S M] :
is_central_scalar S ↥p
source
@[protected]
theorem submodule.nonempty {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
↑p.nonempty
source
@[simp]
theorem submodule.mk_eq_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {x : M} (h : x ∈ p) :
⟨x, h⟩ = 0 ↔ x = 0
source
@[simp, norm_cast]
theorem submodule.coe_eq_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} {x : ↥p} :
↑x = 0 ↔ x = 0
source
@[simp, norm_cast]
theorem submodule.coe_add {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} (x y : ↥p) :
↑(x + y) = ↑x + ↑y
source
@[simp, norm_cast]
theorem submodule.coe_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} :
↑0 = 0
source
@[norm_cast]
theorem submodule.coe_smul {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} (r : R) (x : ↥p) :
↑(r • x) = r • ↑x
source
@[simp, norm_cast]
theorem submodule.coe_smul_of_tower {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} [has_smul S R] [has_smul S M] [is_scalar_tower S R M] (r : S) (x : ↥p) :
↑(r • x) = r • ↑x
source
@[simp, norm_cast]
theorem submodule.coe_mk {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} (x : M) (hx : x ∈ p) :
↑⟨x, hx⟩ = x
source
@[simp]
theorem submodule.coe_mem {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} (x : ↥p) :
↑x ∈ p
source
@[protected, instance]
def submodule.add_comm_monoid {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
add_comm_monoid ↥p
Equations
source
@[protected, instance]
def submodule.module' {S : Type u'} {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) [semiring S] [has_smul S R] [module S M] [is_scalar_tower S R M] :
module S ↥p
Equations
source
@[protected, instance]
def submodule.module {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
module R ↥p
Equations
source
@[protected, instance]
def submodule.no_zero_smul_divisors {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) [no_zero_smul_divisors R M] :
no_zero_smul_divisors R ↥p
source
@[protected]
def submodule.subtype {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
↥p →ₗ[R] M

Embedding of a submodule p to the ambient space M.

Equations
source
theorem submodule.subtype_apply {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) (x : ↥p) :
⇑(p.subtype) x = ↑x
source
@[simp]
theorem submodule.coe_subtype {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
⇑(p.subtype) = coe
source
theorem submodule.injective_subtype {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) :
function.injective ⇑(p.subtype)
source
@[simp]
theorem submodule.coe_sum {R : Type u} {M : Type v} {ι : Type w} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) (x : ι → ↥p) (s : finset ι) :
↑(s.sum (λ (i : ι), x i)) = s.sum (λ (i : ι), ↑(x i))

Note the add_submonoid version of this lemma is called add_submonoid.coe_finset_sum.

Additive actions by submodules #

These instances transfer the action by an element m : M of a R-module M written as m +ᵥ a onto the action by an element s : S of a submodule S : submodule R M such that s +ᵥ a = (s : M) +ᵥ a.

These instances work particularly well in conjunction with add_group.to_add_action, enabling s +ᵥ m as an alias for ↑s + m.

source
@[protected, instance]
def submodule.has_vadd {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {α : Type u_1} [has_vadd M α] :
has_vadd ↥p α
Equations
source
@[protected, instance]
def submodule.vadd_comm_class {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {α : Type u_1} {β : Type u_2} [has_vadd M β] [has_vadd α β] [vadd_comm_class M α β] :
vadd_comm_class ↥p α β
source
@[protected, instance]
def submodule.has_faithful_vadd {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {α : Type u_1} [has_vadd M α] [has_faithful_vadd M α] :
has_faithful_vadd ↥p α
source
@[protected, instance]
def submodule.add_action {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} (p : submodule R M) {α : Type u_1} [add_action M α] :
add_action ↥p α

The action by a submodule is the action by the underlying module.

Equations
source
theorem submodule.vadd_def {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] {module_M : module R M} {p : submodule R M} {α : Type u_1} [has_vadd M α] (g : ↥p) (m : α) :
g +ᵥ m = ↑g +ᵥ m
source
def submodule.restrict_scalars (S : Type u') {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (V : submodule R M) :
submodule S M

V.restrict_scalars S is the S-submodule of the S-module given by restriction of scalars, corresponding to V, an R-submodule of the original R-module.

Equations
Instances for ↥submodule.restrict_scalars
source
@[simp]
theorem submodule.coe_restrict_scalars (S : Type u') {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (V : submodule R M) :
↑(submodule.restrict_scalars S V) = ↑V
source
@[simp]
theorem submodule.restrict_scalars_mem (S : Type u') {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (V : submodule R M) (m : M) :
m ∈ submodule.restrict_scalars S V ↔ m ∈ V
source
@[simp]
theorem submodule.restrict_scalars_self {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (V : submodule R M) :
submodule.restrict_scalars R V = V
source
theorem submodule.restrict_scalars_injective (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] :
function.injective (submodule.restrict_scalars S)
source
@[simp]
theorem submodule.restrict_scalars_inj (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] {V₁ V₂ : submodule R M} :
submodule.restrict_scalars S V₁ = submodule.restrict_scalars S V₂ ↔ V₁ = V₂
source
@[protected, instance]
def submodule.restrict_scalars.orig_module (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (p : submodule R M) :
module R ↥(submodule.restrict_scalars S p)

Even though p.restrict_scalars S has type submodule S M, it is still an R-module.

Equations
source
@[protected, instance]
def submodule.restrict_scalars.is_scalar_tower (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (p : submodule R M) :
is_scalar_tower S R ↥(submodule.restrict_scalars S p)
source
def submodule.restrict_scalars_embedding (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] :
submodule R M ↪o submodule S M

restrict_scalars S is an embedding of the lattice of R-submodules into the lattice of S-submodules.

Equations
source
@[simp]
theorem submodule.restrict_scalars_embedding_apply (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (V : submodule R M) :
⇑(submodule.restrict_scalars_embedding S R M) V = submodule.restrict_scalars S V
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def submodule.restrict_scalars_equiv (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (p : submodule R M) :
↥(submodule.restrict_scalars S p) ≃ₗ[R] ↥p

Turning p : submodule R M into an S-submodule gives the same module structure as turning it into a type and adding a module structure.

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@[simp]
theorem submodule.restrict_scalars_equiv_apply (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (p : submodule R M) (a : ↥(submodule.restrict_scalars S p)) :
⇑(submodule.restrict_scalars_equiv S R M p) a = a
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@[simp]
theorem submodule.restrict_scalars_equiv_symm_apply (S : Type u') (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semiring S] [module S M] [module R M] [has_smul S R] [is_scalar_tower S R M] (p : submodule R M) (a : ↥p) :
⇑((submodule.restrict_scalars_equiv S R M p).symm) a = a
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@[protected, instance]
def submodule.add_subgroup_class {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] :
add_subgroup_class (submodule R M) M
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@[protected]
theorem submodule.neg_mem {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x : M} (hx : x ∈ p) :
-x ∈ p
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def submodule.to_add_subgroup {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) :
add_subgroup M

Reinterpret a submodule as an additive subgroup.

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@[simp]
theorem submodule.coe_to_add_subgroup {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) :
↑(p.to_add_subgroup) = ↑p
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@[simp]
theorem submodule.mem_to_add_subgroup {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x : M} :
x ∈ p.to_add_subgroup ↔ x ∈ p
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theorem submodule.to_add_subgroup_injective {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} :
function.injective submodule.to_add_subgroup
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@[simp]
theorem submodule.to_add_subgroup_eq {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p p' : submodule R M) :
p.to_add_subgroup = p'.to_add_subgroup ↔ p = p'
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theorem submodule.to_add_subgroup_strict_mono {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} :
strict_mono submodule.to_add_subgroup
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theorem submodule.to_add_subgroup_le {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p p' : submodule R M) :
p.to_add_subgroup ≤ p'.to_add_subgroup ↔ p ≤ p'
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theorem submodule.to_add_subgroup_mono {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} :
monotone submodule.to_add_subgroup
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@[protected]
theorem submodule.sub_mem {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x y : M} :
x ∈ p → y ∈ p → x - y ∈ p
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@[protected]
theorem submodule.neg_mem_iff {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x : M} :
-x ∈ p ↔ x ∈ p
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@[protected]
theorem submodule.add_mem_iff_left {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x y : M} :
y ∈ p → (x + y ∈ p ↔ x ∈ p)
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@[protected]
theorem submodule.add_mem_iff_right {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x y : M} :
x ∈ p → (x + y ∈ p ↔ y ∈ p)
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@[protected]
theorem submodule.coe_neg {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) (x : ↥p) :
↑-x = -↑x
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@[protected]
theorem submodule.coe_sub {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) (x y : ↥p) :
↑(x - y) = ↑x - ↑y
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theorem submodule.sub_mem_iff_left {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x y : M} (hy : y ∈ p) :
x - y ∈ p ↔ x ∈ p
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theorem submodule.sub_mem_iff_right {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) {x y : M} (hx : x ∈ p) :
x - y ∈ p ↔ y ∈ p
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@[protected, instance]
def submodule.add_comm_group {R : Type u} {M : Type v} [ring R] [add_comm_group M] {module_M : module R M} (p : submodule R M) :
add_comm_group ↥p
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theorem submodule.not_mem_of_ortho {R : Type u} {M : Type v} [ring R] [is_domain R] [add_comm_group M] [module R M] {x : M} {N : submodule R M} (ortho : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0) :
x ∉ N
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theorem submodule.ne_zero_of_ortho {R : Type u} {M : Type v} [ring R] [is_domain R] [add_comm_group M] [module R M] {x : M} {N : submodule R M} (ortho : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0) :
x ≠ 0
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@[protected, instance]
def submodule.to_ordered_add_comm_monoid {R : Type u} [semiring R] {M : Type u_1} [ordered_add_comm_monoid M] [module R M] (S : submodule R M) :
ordered_add_comm_monoid ↥S

A submodule of an ordered_add_comm_monoid is an ordered_add_comm_monoid.

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@[protected, instance]
def submodule.to_linear_ordered_add_comm_monoid {R : Type u} [semiring R] {M : Type u_1} [linear_ordered_add_comm_monoid M] [module R M] (S : submodule R M) :
linear_ordered_add_comm_monoid ↥S

A submodule of a linear_ordered_add_comm_monoid is a linear_ordered_add_comm_monoid.

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@[protected, instance]
def submodule.to_ordered_cancel_add_comm_monoid {R : Type u} [semiring R] {M : Type u_1} [ordered_cancel_add_comm_monoid M] [module R M] (S : submodule R M) :
ordered_cancel_add_comm_monoid ↥S

A submodule of an ordered_cancel_add_comm_monoid is an ordered_cancel_add_comm_monoid.

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@[protected, instance]
def submodule.to_linear_ordered_cancel_add_comm_monoid {R : Type u} [semiring R] {M : Type u_1} [linear_ordered_cancel_add_comm_monoid M] [module R M] (S : submodule R M) :
linear_ordered_cancel_add_comm_monoid ↥S

A submodule of a linear_ordered_cancel_add_comm_monoid is a linear_ordered_cancel_add_comm_monoid.

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@[protected, instance]
def submodule.to_ordered_add_comm_group {R : Type u} [ring R] {M : Type u_1} [ordered_add_comm_group M] [module R M] (S : submodule R M) :
ordered_add_comm_group ↥S

A submodule of an ordered_add_comm_group is an ordered_add_comm_group.

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@[protected, instance]
def submodule.to_linear_ordered_add_comm_group {R : Type u} [ring R] {M : Type u_1} [linear_ordered_add_comm_group M] [module R M] (S : submodule R M) :
linear_ordered_add_comm_group ↥S

A submodule of a linear_ordered_add_comm_group is a linear_ordered_add_comm_group.

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theorem submodule.smul_mem_iff {S : Type u'} {R : Type u} {M : Type v} [division_ring S] [semiring R] [add_comm_monoid M] [module R M] [has_smul S R] [module S M] [is_scalar_tower S R M] (p : submodule R M) {s : S} {x : M} (s0 : s ≠ 0) :
s • x ∈ p ↔ x ∈ p
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@[reducible]
def subspace (R : Type u) (M : Type v) [division_ring R] [add_comm_group M] [module R M] :
Type v

Subspace of a vector space. Defined to equal submodule.