Pointwise operations of sets in a group with zero #
This file proves properties of pointwise operations of sets in a group with zero.
Tags #
set multiplication, set addition, pointwise addition, pointwise multiplication, pointwise subtraction
def
Set.smulZeroClassSet
{α : Type u_1}
{β : Type u_2}
[Zero β]
[SMulZeroClass α β]
:
SMulZeroClass α (Set β)
If scalar multiplication by elements of α
sends (0 : β)
to zero,
then the same is true for (0 : Set β)
.
Equations
- Set.smulZeroClassSet = { toSMul := Set.smulSet, smul_zero := ⋯ }
Instances For
theorem
Set.smul_zero_subset
{α : Type u_1}
{β : Type u_2}
[Zero β]
[SMulZeroClass α β]
(s : Set α)
:
theorem
Set.Nonempty.smul_zero
{α : Type u_1}
{β : Type u_2}
[Zero β]
[SMulZeroClass α β]
{s : Set α}
(hs : s.Nonempty)
:
theorem
Set.zero_mem_smul_set
{α : Type u_1}
{β : Type u_2}
[Zero β]
[SMulZeroClass α β]
{t : Set β}
{a : α}
(h : 0 ∈ t)
:
theorem
Set.zero_mem_smul_set_iff
{α : Type u_1}
{β : Type u_2}
[Zero β]
[SMulZeroClass α β]
{t : Set β}
{a : α}
[Zero α]
[NoZeroSMulDivisors α β]
(ha : a ≠ 0)
:
Note that we have neither SMulWithZero α (Set β)
nor SMulWithZero (Set α) (Set β)
because 0 * ∅ ≠ 0
.
theorem
Set.zero_smul_subset
{α : Type u_1}
{β : Type u_2}
[Zero α]
[Zero β]
[SMulWithZero α β]
(t : Set β)
:
theorem
Set.zero_smul_set_subset
{α : Type u_1}
{β : Type u_2}
[Zero α]
[Zero β]
[SMulWithZero α β]
(s : Set β)
:
theorem
Set.subsingleton_zero_smul_set
{α : Type u_1}
{β : Type u_2}
[Zero α]
[Zero β]
[SMulWithZero α β]
(s : Set β)
:
(0 • s).Subsingleton
noncomputable def
Set.distribSMulSet
{α : Type u_1}
{β : Type u_2}
[AddZeroClass β]
[DistribSMul α β]
:
DistribSMul α (Set β)
If the scalar multiplication (· • ·) : α → β → β
is distributive,
then so is (· • ·) : α → Set β → Set β
.
Equations
- Set.distribSMulSet = { toSMulZeroClass := Set.smulZeroClassSet, smul_add := ⋯ }
Instances For
noncomputable def
Set.distribMulActionSet
{α : Type u_1}
{β : Type u_2}
[Monoid α]
[AddMonoid β]
[DistribMulAction α β]
:
DistribMulAction α (Set β)
A distributive multiplicative action of a monoid on an additive monoid β
gives a distributive
multiplicative action on Set β
.
Equations
- Set.distribMulActionSet = { toMulAction := Set.mulActionSet, smul_zero := ⋯, smul_add := ⋯ }
Instances For
noncomputable def
Set.mulDistribMulActionSet
{α : Type u_1}
{β : Type u_2}
[Monoid α]
[Monoid β]
[MulDistribMulAction α β]
:
MulDistribMulAction α (Set β)
A multiplicative action of a monoid on a monoid β
gives a multiplicative action on Set β
.
Equations
- Set.mulDistribMulActionSet = { toMulAction := Set.mulActionSet, smul_mul := ⋯, smul_one := ⋯ }
Instances For
instance
Set.instNoZeroSMulDivisors
{α : Type u_1}
{β : Type u_2}
[Zero α]
[Zero β]
[SMul α β]
[NoZeroSMulDivisors α β]
:
NoZeroSMulDivisors (Set α) (Set β)
instance
Set.noZeroSMulDivisors_set
{α : Type u_1}
{β : Type u_2}
[Zero α]
[Zero β]
[SMul α β]
[NoZeroSMulDivisors α β]
:
NoZeroSMulDivisors α (Set β)
instance
Set.instNoZeroDivisors
{α : Type u_1}
[Zero α]
[Mul α]
[NoZeroDivisors α]
:
NoZeroDivisors (Set α)
theorem
Set.smul_univ₀'
{α : Type u_1}
{β : Type u_2}
[GroupWithZero α]
[MulAction α β]
{s : Set α}
(hs : s.Nontrivial)
:
@[simp]
@[simp]