mathlib documentation

group_theory.group_action.sub_mul_action

Sets invariant to a mul_action #

In this file we define sub_mul_action R M; a subset of a mul_action R M which is closed with respect to scalar multiplication.

For most uses, typically submodule R M is more powerful.

Main definitions #

Tags #

submodule, mul_action

structure sub_mul_action (R : Type u) (M : Type v) [has_scalar R M] :
Type v

A sub_mul_action is a set which is closed under scalar multiplication.

@[instance]
def sub_mul_action.set_like {R : Type u} {M : Type v} [has_scalar R M] :
Equations
@[simp]
theorem sub_mul_action.mem_carrier {R : Type u} {M : Type v} [has_scalar R M] {p : sub_mul_action R M} {x : M} :
@[ext]
theorem sub_mul_action.ext {R : Type u} {M : Type v} [has_scalar R M] {p q : sub_mul_action R M} (h : ∀ (x : M), x p x q) :
p = q
def sub_mul_action.copy {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) (s : set M) (hs : s = p) :

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

Equations
@[simp]
theorem sub_mul_action.coe_copy {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) (s : set M) (hs : s = p) :
(p.copy s hs) = s
theorem sub_mul_action.copy_eq {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) (s : set M) (hs : s = p) :
p.copy s hs = p
@[instance]
def sub_mul_action.has_bot {R : Type u} {M : Type v} [has_scalar R M] :
Equations
@[instance]
def sub_mul_action.inhabited {R : Type u} {M : Type v} [has_scalar R M] :
Equations
theorem sub_mul_action.smul_mem {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) {x : M} (r : R) (h : x p) :
r x p
@[instance]
def sub_mul_action.has_scalar {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) :
Equations
@[simp]
theorem sub_mul_action.coe_smul {R : Type u} {M : Type v} [has_scalar R M] {p : sub_mul_action R M} (r : R) (x : p) :
(r x) = r x
@[simp]
theorem sub_mul_action.coe_mk {R : Type u} {M : Type v} [has_scalar R M] {p : sub_mul_action R M} (x : M) (hx : x p) :
x, hx⟩ = x
def sub_mul_action.subtype {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) :

Embedding of a submodule p to the ambient space M.

Equations
@[simp]
theorem sub_mul_action.subtype_apply {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) (x : p) :
theorem sub_mul_action.subtype_eq_val {R : Type u} {M : Type v} [has_scalar R M] (p : sub_mul_action R M) :
theorem sub_mul_action.smul_of_tower_mem {S : Type u'} {R : Type u} {M : Type v} [monoid R] [mul_action R M] [has_scalar S R] [has_scalar S M] [is_scalar_tower S R M] (p : sub_mul_action R M) (s : S) {x : M} (h : x p) :
s x p
@[instance]
def sub_mul_action.has_scalar' {S : Type u'} {R : Type u} {M : Type v} [monoid R] [mul_action R M] [has_scalar S R] [has_scalar S M] [is_scalar_tower S R M] (p : sub_mul_action R M) :
Equations
@[instance]
def sub_mul_action.is_scalar_tower {S : Type u'} {R : Type u} {M : Type v} [monoid R] [mul_action R M] [has_scalar S R] [has_scalar S M] [is_scalar_tower S R M] (p : sub_mul_action R M) :
@[simp]
theorem sub_mul_action.coe_smul_of_tower {S : Type u'} {R : Type u} {M : Type v} [monoid R] [mul_action R M] [has_scalar S R] [has_scalar S M] [is_scalar_tower S R M] (p : sub_mul_action R M) (s : S) (x : p) :
(s x) = s x
@[simp]
theorem sub_mul_action.smul_mem_iff' {R : Type u} {M : Type v} [monoid R] [mul_action R M] (p : sub_mul_action R M) {G : Type u_1} [group G] [has_scalar G R] [mul_action G M] [is_scalar_tower G R M] (g : G) {x : M} :
g x p x p
@[instance]
def sub_mul_action.mul_action' {S : Type u'} {R : Type u} {M : Type v} [monoid R] [mul_action R M] [monoid S] [has_scalar S R] [mul_action S M] [is_scalar_tower S R M] (p : sub_mul_action R M) :

If the scalar product forms a mul_action, then the subset inherits this action

Equations
@[instance]
def sub_mul_action.mul_action {R : Type u} {M : Type v} [monoid R] [mul_action R M] (p : sub_mul_action R M) :
Equations
theorem sub_mul_action.zero_mem {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (p : sub_mul_action R M) (h : p.nonempty) :
0 p
@[instance]
def sub_mul_action.has_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [module R M] (p : sub_mul_action R M) [n_empty : nonempty p] :

If the scalar product forms a module, and the sub_mul_action is not , then the subset inherits the zero.

Equations
theorem sub_mul_action.neg_mem {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] (p : sub_mul_action R M) {x : M} (hx : x p) :
-x p
@[simp]
theorem sub_mul_action.neg_mem_iff {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] (p : sub_mul_action R M) {x : M} :
-x p x p
@[instance]
def sub_mul_action.has_neg {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] (p : sub_mul_action R M) :
Equations
@[simp]
theorem sub_mul_action.coe_neg {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] (p : sub_mul_action R M) (x : p) :
theorem sub_mul_action.smul_mem_iff {S : Type u'} {R : Type u} {M : Type v} [division_ring S] [semiring R] [mul_action R M] [has_scalar S R] [mul_action S M] [is_scalar_tower S R M] (p : sub_mul_action R M) {s : S} {x : M} (s0 : s 0) :
s x p x p