# mathlibdocumentation

data.set.pointwise

# Pointwise operations of sets #

This file defines pointwise algebraic operations on sets.

## Main declarations #

For sets `s` and `t` and scalar `a`:

• `s * t`: Multiplication, set of all `x * y` where `x ∈ s` and `y ∈ t`.
• `s + t`: Addition, set of all `x + y` where `x ∈ s` and `y ∈ t`.
• `s⁻¹`: Inversion, set of all `x⁻¹` where `x ∈ s`.
• `-s`: Negation, set of all `-x` where `x ∈ s`.
• `s / t`: Division, set of all `x / y` where `x ∈ s` and `y ∈ t`.
• `s - t`: Subtraction, set of all `x - y` where `x ∈ s` and `y ∈ t`.
• `s • t`: Scalar multiplication, set of all `x • y` where `x ∈ s` and `y ∈ t`.
• `s +ᵥ t`: Scalar addition, set of all `x +ᵥ y` where `x ∈ s` and `y ∈ t`.
• `s -ᵥ t`: Scalar subtraction, set of all `x -ᵥ y` where `x ∈ s` and `y ∈ t`.
• `a • s`: Scaling, set of all `a • x` where `x ∈ s`.
• `a +ᵥ s`: Translation, set of all `a +ᵥ x` where `x ∈ s`.

For `α` a semigroup/monoid, `set α` is a semigroup/monoid. As an unfortunate side effect, this means that `n • s`, where `n : ℕ`, is ambiguous between pointwise scaling and repeated pointwise addition; the former has `(2 : ℕ) • {1, 2} = {2, 4}`, while the latter has `(2 : ℕ) • {1, 2} = {2, 3, 4}`. See note [pointwise nat action].

Appropriate definitions and results are also transported to the additive theory via `to_additive`.

### Definitions for Hahn series #

• `set.add_antidiagonal s t a`, `set.mul_antidiagonal s t a`: Sets of pairs of elements of `s` and `t` that add/multiply to `a`.
• `finset.add_antidiagonal`, `finset.mul_antidiagonal`: Finset versions of the above when `s` and `t` are well-founded.

## Implementation notes #

• The following expressions are considered in simp-normal form in a group: `(λ h, h * g) ⁻¹' s`, `(λ h, g * h) ⁻¹' s`, `(λ h, h * g⁻¹) ⁻¹' s`, `(λ h, g⁻¹ * h) ⁻¹' s`, `s * t`, `s⁻¹`, `(1 : set _)` (and similarly for additive variants). Expressions equal to one of these will be simplified.
• We put all instances in the locale `pointwise`, so that these instances are not available by default. Note that we do not mark them as reducible (as argued by note [reducible non-instances]) since we expect the locale to be open whenever the instances are actually used (and making the instances reducible changes the behavior of `simp`.

## Tags #

set multiplication, set addition, pointwise addition, pointwise multiplication, pointwise subtraction

Pointwise monoids (`set`, `finset`, `filter`) have derived pointwise actions of the form `has_smul α β → has_smul α (set β)`. When `α` is `ℕ` or `ℤ`, this action conflicts with the nat or int action coming from `set β` being a `monoid` or `div_inv_monoid`. For example, `2 • {a, b}` can both be `{2 • a, 2 • b}` (pointwise action, pointwise repeated addition, `set.has_smul_set`) and `{a + a, a + b, b + a, b + b}` (nat or int action, repeated pointwise addition, `set.has_nsmul`).

Because the pointwise action can easily be spelled out in such cases, we give higher priority to the nat and int actions.

### `0`/`1` as sets #

@[protected]
def set.has_one {α : Type u_2} [has_one α] :

The set `1 : set α` is defined as `{1}` in locale `pointwise`.

Equations
@[protected]
def set.has_zero {α : Type u_2} [has_zero α] :

The set `0 : set α` is defined as `{0}` in locale `pointwise`.

Equations
theorem set.singleton_one {α : Type u_2} [has_one α] :
{1} = 1
theorem set.singleton_zero {α : Type u_2} [has_zero α] :
{0} = 0
@[simp]
theorem set.mem_one {α : Type u_2} [has_one α] {a : α} :
a 1 a = 1
@[simp]
theorem set.mem_zero {α : Type u_2} [has_zero α] {a : α} :
a 0 a = 0
theorem set.zero_mem_zero {α : Type u_2} [has_zero α] :
0 0
theorem set.one_mem_one {α : Type u_2} [has_one α] :
1 1
@[simp]
theorem set.zero_subset {α : Type u_2} [has_zero α] {s : set α} :
0 s 0 s
@[simp]
theorem set.one_subset {α : Type u_2} [has_one α] {s : set α} :
1 s 1 s
theorem set.zero_nonempty {α : Type u_2} [has_zero α] :
theorem set.one_nonempty {α : Type u_2} [has_one α] :
@[simp]
theorem set.image_zero {α : Type u_2} {β : Type u_3} [has_zero α] {f : α → β} :
f '' 0 = {f 0}
@[simp]
theorem set.image_one {α : Type u_2} {β : Type u_3} [has_one α] {f : α → β} :
f '' 1 = {f 1}
theorem set.subset_zero_iff_eq {α : Type u_2} [has_zero α] {s : set α} :
s 0 s = s = 0
theorem set.subset_one_iff_eq {α : Type u_2} [has_one α] {s : set α} :
s 1 s = s = 1
theorem set.nonempty.subset_zero_iff {α : Type u_2} [has_zero α] {s : set α} (h : s.nonempty) :
s 0 s = 0
theorem set.nonempty.subset_one_iff {α : Type u_2} [has_one α] {s : set α} (h : s.nonempty) :
s 1 s = 1
def set.singleton_zero_hom {α : Type u_2} [has_zero α] :
(set α)

The singleton operation as a `zero_hom`.

Equations
def set.singleton_one_hom {α : Type u_2} [has_one α] :
(set α)

The singleton operation as a `one_hom`.

Equations
@[simp]
theorem set.coe_singleton_zero_hom {α : Type u_2} [has_zero α] :
@[simp]
theorem set.coe_singleton_one_hom {α : Type u_2} [has_one α] :

### Set negation/inversion #

@[protected]
def set.has_neg {α : Type u_2} [has_neg α] :

The pointwise negation of set `-s` is defined as `{x | -x ∈ s}` in locale `pointwise`. It is equal to `{-x | x ∈ s}`, see `set.image_neg`.

Equations
@[protected]
def set.has_inv {α : Type u_2} [has_inv α] :

The pointwise inversion of set `s⁻¹` is defined as `{x | x⁻¹ ∈ s}` in locale `pointwise`. It i equal to `{x⁻¹ | x ∈ s}`, see `set.image_inv`.

Equations
@[simp]
theorem set.mem_inv {α : Type u_2} [has_inv α] {s : set α} {a : α} :
@[simp]
theorem set.mem_neg {α : Type u_2} [has_neg α] {s : set α} {a : α} :
a -s -a s
@[simp]
theorem set.inv_preimage {α : Type u_2} [has_inv α] {s : set α} :
@[simp]
theorem set.neg_preimage {α : Type u_2} [has_neg α] {s : set α} :
= -s
@[simp]
theorem set.neg_empty {α : Type u_2} [has_neg α] :
@[simp]
theorem set.inv_empty {α : Type u_2} [has_inv α] :
@[simp]
theorem set.neg_univ {α : Type u_2} [has_neg α] :
@[simp]
theorem set.inv_univ {α : Type u_2} [has_inv α] :
@[simp]
theorem set.inter_neg {α : Type u_2} [has_neg α] {s t : set α} :
-(s t) = -s -t
@[simp]
theorem set.inter_inv {α : Type u_2} [has_inv α] {s t : set α} :
@[simp]
theorem set.union_neg {α : Type u_2} [has_neg α] {s t : set α} :
-(s t) = -s -t
@[simp]
theorem set.union_inv {α : Type u_2} [has_inv α] {s t : set α} :
@[simp]
theorem set.Inter_inv {α : Type u_2} {ι : Sort u_5} [has_inv α] (s : ι → set α) :
(⋂ (i : ι), s i)⁻¹ = ⋂ (i : ι), (s i)⁻¹
@[simp]
theorem set.Inter_neg {α : Type u_2} {ι : Sort u_5} [has_neg α] (s : ι → set α) :
(-⋂ (i : ι), s i) = ⋂ (i : ι), -s i
@[simp]
theorem set.Union_inv {α : Type u_2} {ι : Sort u_5} [has_inv α] (s : ι → set α) :
(⋃ (i : ι), s i)⁻¹ = ⋃ (i : ι), (s i)⁻¹
@[simp]
theorem set.Union_neg {α : Type u_2} {ι : Sort u_5} [has_neg α] (s : ι → set α) :
(-⋃ (i : ι), s i) = ⋃ (i : ι), -s i
@[simp]
theorem set.compl_neg {α : Type u_2} [has_neg α] {s : set α} :
-s = (-s)
@[simp]
theorem set.compl_inv {α : Type u_2} [has_inv α] {s : set α} :
theorem set.inv_mem_inv {α : Type u_2} {s : set α} {a : α} :
theorem set.neg_mem_neg {α : Type u_2} {s : set α} {a : α} :
-a -s a s
@[simp]
theorem set.nonempty_neg {α : Type u_2} {s : set α} :
@[simp]
theorem set.nonempty_inv {α : Type u_2} {s : set α} :
theorem set.nonempty.inv {α : Type u_2} {s : set α} (h : s.nonempty) :
theorem set.nonempty.neg {α : Type u_2} {s : set α} (h : s.nonempty) :
theorem set.finite.inv {α : Type u_2} {s : set α} (hs : s.finite) :
theorem set.finite.neg {α : Type u_2} {s : set α} (hs : s.finite) :
@[simp]
theorem set.image_inv {α : Type u_2} {s : set α} :
@[simp]
theorem set.image_neg {α : Type u_2} {s : set α} :
= -s
@[protected, simp, instance]
def set.has_involutive_neg {α : Type u_2}  :
Equations
@[protected, simp, instance]
def set.has_involutive_inv {α : Type u_2}  :
Equations
@[simp]
theorem set.neg_subset_neg {α : Type u_2} {s t : set α} :
-s -t s t
@[simp]
theorem set.inv_subset_inv {α : Type u_2} {s t : set α} :
theorem set.inv_subset {α : Type u_2} {s t : set α} :
theorem set.neg_subset {α : Type u_2} {s t : set α} :
-s t s -t
@[simp]
theorem set.inv_singleton {α : Type u_2} (a : α) :
{a}⁻¹ = {a⁻¹}
@[simp]
theorem set.neg_singleton {α : Type u_2} (a : α) :
-{a} = {-a}
theorem set.inv_range {α : Type u_2} {ι : Sort u_1} {f : ι → α} :
(set.range f)⁻¹ = set.range (λ (i : ι), (f i)⁻¹)
theorem set.neg_range {α : Type u_2} {ι : Sort u_1} {f : ι → α} :
= set.range (λ (i : ι), -f i)
theorem set.image_op_neg {α : Type u_2} {s : set α} :
theorem set.image_op_inv {α : Type u_2} {s : set α} :

### Set addition/multiplication #

@[protected]
def set.has_mul {α : Type u_2} [has_mul α] :

The pointwise multiplication of sets `s * t` and `t` is defined as `{x * y | x ∈ s, y ∈ t}` in locale `pointwise`.

Equations
@[protected]
def set.has_add {α : Type u_2} [has_add α] :

The pointwise addition of sets `s + t` is defined as `{x + y | x ∈ s, y ∈ t}` in locale `pointwise`.

Equations
@[simp]
theorem set.image2_mul {α : Type u_2} [has_mul α] {s t : set α} :
= s * t
@[simp]
theorem set.image2_add {α : Type u_2} [has_add α] {s t : set α} :
= s + t
theorem set.mem_add {α : Type u_2} [has_add α] {s t : set α} {a : α} :
a s + t ∃ (x y : α), x s y t x + y = a
theorem set.mem_mul {α : Type u_2} [has_mul α] {s t : set α} {a : α} :
a s * t ∃ (x y : α), x s y t x * y = a
theorem set.mul_mem_mul {α : Type u_2} [has_mul α] {s t : set α} {a b : α} :
a sb ta * b s * t
theorem set.add_mem_add {α : Type u_2} [has_add α] {s t : set α} {a b : α} :
a sb ta + b s + t
theorem set.image_mul_prod {α : Type u_2} [has_mul α] {s t : set α} :
(λ (x : α × α), x.fst * x.snd) '' s ×ˢ t = s * t
theorem set.add_image_prod {α : Type u_2} [has_add α] {s t : set α} :
(λ (x : α × α), x.fst + x.snd) '' s ×ˢ t = s + t
@[simp]
theorem set.empty_mul {α : Type u_2} [has_mul α] {s : set α} :
@[simp]
theorem set.empty_add {α : Type u_2} [has_add α] {s : set α} :
@[simp]
theorem set.add_empty {α : Type u_2} [has_add α] {s : set α} :
@[simp]
theorem set.mul_empty {α : Type u_2} [has_mul α] {s : set α} :
@[simp]
theorem set.add_eq_empty {α : Type u_2} [has_add α] {s t : set α} :
s + t = s = t =
@[simp]
theorem set.mul_eq_empty {α : Type u_2} [has_mul α] {s t : set α} :
s * t = s = t =
@[simp]
theorem set.mul_nonempty {α : Type u_2} [has_mul α] {s t : set α} :
(s * t).nonempty
@[simp]
theorem set.add_nonempty {α : Type u_2} [has_add α] {s t : set α} :
(s + t).nonempty
theorem set.nonempty.add {α : Type u_2} [has_add α] {s t : set α} :
s.nonemptyt.nonempty(s + t).nonempty
theorem set.nonempty.mul {α : Type u_2} [has_mul α] {s t : set α} :
s.nonemptyt.nonempty(s * t).nonempty
theorem set.nonempty.of_add_left {α : Type u_2} [has_add α] {s t : set α} :
(s + t).nonempty → s.nonempty
theorem set.nonempty.of_mul_left {α : Type u_2} [has_mul α] {s t : set α} :
(s * t).nonempty → s.nonempty
theorem set.nonempty.of_add_right {α : Type u_2} [has_add α] {s t : set α} :
(s + t).nonempty → t.nonempty
theorem set.nonempty.of_mul_right {α : Type u_2} [has_mul α] {s t : set α} :
(s * t).nonempty → t.nonempty
@[simp]
theorem set.add_singleton {α : Type u_2} [has_add α] {s : set α} {b : α} :
s + {b} = (λ (_x : α), _x + b) '' s
@[simp]
theorem set.mul_singleton {α : Type u_2} [has_mul α] {s : set α} {b : α} :
s * {b} = (λ (_x : α), _x * b) '' s
@[simp]
theorem set.singleton_add {α : Type u_2} [has_add α] {t : set α} {a : α} :
{a} + t =
@[simp]
theorem set.singleton_mul {α : Type u_2} [has_mul α] {t : set α} {a : α} :
{a} * t =
@[simp]
theorem set.singleton_mul_singleton {α : Type u_2} [has_mul α] {a b : α} :
{a} * {b} = {a * b}
@[simp]
theorem set.singleton_add_singleton {α : Type u_2} [has_add α] {a b : α} :
{a} + {b} = {a + b}
theorem set.add_subset_add {α : Type u_2} [has_add α] {s₁ s₂ t₁ t₂ : set α} :
s₁ t₁s₂ t₂s₁ + s₂ t₁ + t₂
theorem set.mul_subset_mul {α : Type u_2} [has_mul α] {s₁ s₂ t₁ t₂ : set α} :
s₁ t₁s₂ t₂s₁ * s₂ t₁ * t₂
theorem set.add_subset_add_left {α : Type u_2} [has_add α] {s t₁ t₂ : set α} :
t₁ t₂s + t₁ s + t₂
theorem set.mul_subset_mul_left {α : Type u_2} [has_mul α] {s t₁ t₂ : set α} :
t₁ t₂s * t₁ s * t₂
theorem set.mul_subset_mul_right {α : Type u_2} [has_mul α] {s₁ s₂ t : set α} :
s₁ s₂s₁ * t s₂ * t
theorem set.add_subset_add_right {α : Type u_2} [has_add α] {s₁ s₂ t : set α} :
s₁ s₂s₁ + t s₂ + t
theorem set.mul_subset_iff {α : Type u_2} [has_mul α] {s t u : set α} :
s * t u ∀ (x : α), x s∀ (y : α), y tx * y u
theorem set.add_subset_iff {α : Type u_2} [has_add α] {s t u : set α} :
s + t u ∀ (x : α), x s∀ (y : α), y tx + y u
theorem set.union_mul {α : Type u_2} [has_mul α] {s₁ s₂ t : set α} :
(s₁ s₂) * t = s₁ * t s₂ * t
theorem set.union_add {α : Type u_2} [has_add α] {s₁ s₂ t : set α} :
s₁ s₂ + t = s₁ + t (s₂ + t)
theorem set.mul_union {α : Type u_2} [has_mul α] {s t₁ t₂ : set α} :
s * (t₁ t₂) = s * t₁ s * t₂
theorem set.add_union {α : Type u_2} [has_add α] {s t₁ t₂ : set α} :
s + (t₁ t₂) = s + t₁ (s + t₂)
theorem set.inter_add_subset {α : Type u_2} [has_add α] {s₁ s₂ t : set α} :
s₁ s₂ + t (s₁ + t) (s₂ + t)
theorem set.inter_mul_subset {α : Type u_2} [has_mul α] {s₁ s₂ t : set α} :
s₁ s₂ * t s₁ * t (s₂ * t)
theorem set.mul_inter_subset {α : Type u_2} [has_mul α] {s t₁ t₂ : set α} :
s * (t₁ t₂) s * t₁ (s * t₂)
theorem set.add_inter_subset {α : Type u_2} [has_add α] {s t₁ t₂ : set α} :
s + t₁ t₂ (s + t₁) (s + t₂)
theorem set.Union_mul_left_image {α : Type u_2} [has_mul α] {s t : set α} :
(⋃ (a : α) (H : a s), '' t) = s * t
theorem set.Union_add_left_image {α : Type u_2} [has_add α] {s t : set α} :
(⋃ (a : α) (H : a s), '' t) = s + t
theorem set.Union_mul_right_image {α : Type u_2} [has_mul α] {s t : set α} :
(⋃ (a : α) (H : a t), (λ (_x : α), _x * a) '' s) = s * t
theorem set.Union_add_right_image {α : Type u_2} [has_add α] {s t : set α} :
(⋃ (a : α) (H : a t), (λ (_x : α), _x + a) '' s) = s + t
theorem set.Union_add {α : Type u_2} {ι : Sort u_5} [has_add α] (s : ι → set α) (t : set α) :
(⋃ (i : ι), s i) + t = ⋃ (i : ι), s i + t
theorem set.Union_mul {α : Type u_2} {ι : Sort u_5} [has_mul α] (s : ι → set α) (t : set α) :
(⋃ (i : ι), s i) * t = ⋃ (i : ι), s i * t
theorem set.add_Union {α : Type u_2} {ι : Sort u_5} [has_add α] (s : set α) (t : ι → set α) :
(s + ⋃ (i : ι), t i) = ⋃ (i : ι), s + t i
theorem set.mul_Union {α : Type u_2} {ι : Sort u_5} [has_mul α] (s : set α) (t : ι → set α) :
(s * ⋃ (i : ι), t i) = ⋃ (i : ι), s * t i
theorem set.Union₂_add {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_add α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋃ (i : ι) (j : κ i), s i j) + t = ⋃ (i : ι) (j : κ i), s i j + t
theorem set.Union₂_mul {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_mul α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋃ (i : ι) (j : κ i), s i j) * t = ⋃ (i : ι) (j : κ i), s i j * t
theorem set.add_Union₂ {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_add α] (s : set α) (t : Π (i : ι), κ iset α) :
(s + ⋃ (i : ι) (j : κ i), t i j) = ⋃ (i : ι) (j : κ i), s + t i j
theorem set.mul_Union₂ {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_mul α] (s : set α) (t : Π (i : ι), κ iset α) :
(s * ⋃ (i : ι) (j : κ i), t i j) = ⋃ (i : ι) (j : κ i), s * t i j
theorem set.Inter_add_subset {α : Type u_2} {ι : Sort u_5} [has_add α] (s : ι → set α) (t : set α) :
(⋂ (i : ι), s i) + t ⋂ (i : ι), s i + t
theorem set.Inter_mul_subset {α : Type u_2} {ι : Sort u_5} [has_mul α] (s : ι → set α) (t : set α) :
(⋂ (i : ι), s i) * t ⋂ (i : ι), s i * t
theorem set.add_Inter_subset {α : Type u_2} {ι : Sort u_5} [has_add α] (s : set α) (t : ι → set α) :
(s + ⋂ (i : ι), t i) ⋂ (i : ι), s + t i
theorem set.mul_Inter_subset {α : Type u_2} {ι : Sort u_5} [has_mul α] (s : set α) (t : ι → set α) :
(s * ⋂ (i : ι), t i) ⋂ (i : ι), s * t i
theorem set.Inter₂_mul_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_mul α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋂ (i : ι) (j : κ i), s i j) * t ⋂ (i : ι) (j : κ i), s i j * t
theorem set.Inter₂_add_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_add α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋂ (i : ι) (j : κ i), s i j) + t ⋂ (i : ι) (j : κ i), s i j + t
theorem set.add_Inter₂_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_add α] (s : set α) (t : Π (i : ι), κ iset α) :
(s + ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), s + t i j
theorem set.mul_Inter₂_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_mul α] (s : set α) (t : Π (i : ι), κ iset α) :
(s * ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), s * t i j
theorem set.finite.mul {α : Type u_2} [has_mul α] {s t : set α} :
s.finitet.finite(s * t).finite
theorem set.finite.add {α : Type u_2} [has_add α] {s t : set α} :
s.finitet.finite(s + t).finite
def set.fintype_add {α : Type u_2} [has_add α] [decidable_eq α] (s t : set α) [fintype s] [fintype t] :

Equations
def set.fintype_mul {α : Type u_2} [has_mul α] [decidable_eq α] (s t : set α) [fintype s] [fintype t] :

Multiplication preserves finiteness.

Equations
def set.singleton_mul_hom {α : Type u_2} [has_mul α] :
α →ₙ* set α

The singleton operation as a `mul_hom`.

Equations
def set.singleton_add_hom {α : Type u_2} [has_add α] :
(set α)

The singleton operation as an `add_hom`.

Equations
@[simp]
theorem set.coe_singleton_add_hom {α : Type u_2} [has_add α] :
@[simp]
theorem set.coe_singleton_mul_hom {α : Type u_2} [has_mul α] :
@[simp]
theorem set.singleton_add_hom_apply {α : Type u_2} [has_add α] (a : α) :
@[simp]
theorem set.singleton_mul_hom_apply {α : Type u_2} [has_mul α] (a : α) :
@[simp]
theorem set.image_op_add {α : Type u_2} [has_add α] {s t : set α} :
@[simp]
theorem set.image_op_mul {α : Type u_2} [has_mul α] {s t : set α} :

### Set subtraction/division #

@[protected]
def set.has_div {α : Type u_2} [has_div α] :

The pointwise division of sets `s / t` is defined as `{x / y | x ∈ s, y ∈ t}` in locale `pointwise`.

Equations
@[protected]
def set.has_sub {α : Type u_2} [has_sub α] :

The pointwise subtraction of sets `s - t` is defined as `{x - y | x ∈ s, y ∈ t}` in locale `pointwise`.

Equations
@[simp]
theorem set.image2_sub {α : Type u_2} [has_sub α] {s t : set α} :
= s - t
@[simp]
theorem set.image2_div {α : Type u_2} [has_div α] {s t : set α} :
= s / t
theorem set.mem_sub {α : Type u_2} [has_sub α] {s t : set α} {a : α} :
a s - t ∃ (x y : α), x s y t x - y = a
theorem set.mem_div {α : Type u_2} [has_div α] {s t : set α} {a : α} :
a s / t ∃ (x y : α), x s y t x / y = a
theorem set.sub_mem_sub {α : Type u_2} [has_sub α] {s t : set α} {a b : α} :
a sb ta - b s - t
theorem set.div_mem_div {α : Type u_2} [has_div α] {s t : set α} {a b : α} :
a sb ta / b s / t
theorem set.image_div_prod {α : Type u_2} [has_div α] {s t : set α} :
(λ (x : α × α), x.fst / x.snd) '' s ×ˢ t = s / t
@[simp]
theorem set.empty_div {α : Type u_2} [has_div α] {s : set α} :
@[simp]
theorem set.empty_sub {α : Type u_2} [has_sub α] {s : set α} :
@[simp]
theorem set.sub_empty {α : Type u_2} [has_sub α] {s : set α} :
@[simp]
theorem set.div_empty {α : Type u_2} [has_div α] {s : set α} :
@[simp]
theorem set.div_eq_empty {α : Type u_2} [has_div α] {s t : set α} :
s / t = s = t =
@[simp]
theorem set.sub_eq_empty {α : Type u_2} [has_sub α] {s t : set α} :
s - t = s = t =
@[simp]
theorem set.div_nonempty {α : Type u_2} [has_div α] {s t : set α} :
(s / t).nonempty
@[simp]
theorem set.sub_nonempty {α : Type u_2} [has_sub α] {s t : set α} :
(s - t).nonempty
theorem set.nonempty.div {α : Type u_2} [has_div α] {s t : set α} :
s.nonemptyt.nonempty(s / t).nonempty
theorem set.nonempty.sub {α : Type u_2} [has_sub α] {s t : set α} :
s.nonemptyt.nonempty(s - t).nonempty
theorem set.nonempty.of_sub_left {α : Type u_2} [has_sub α] {s t : set α} :
(s - t).nonempty → s.nonempty
theorem set.nonempty.of_div_left {α : Type u_2} [has_div α] {s t : set α} :
(s / t).nonempty → s.nonempty
theorem set.nonempty.of_div_right {α : Type u_2} [has_div α] {s t : set α} :
(s / t).nonempty → t.nonempty
theorem set.nonempty.of_sub_right {α : Type u_2} [has_sub α] {s t : set α} :
(s - t).nonempty → t.nonempty
@[simp]
theorem set.div_singleton {α : Type u_2} [has_div α] {s : set α} {b : α} :
s / {b} = (λ (_x : α), _x / b) '' s
@[simp]
theorem set.sub_singleton {α : Type u_2} [has_sub α] {s : set α} {b : α} :
s - {b} = (λ (_x : α), _x - b) '' s
@[simp]
theorem set.singleton_div {α : Type u_2} [has_div α] {t : set α} {a : α} :
{a} / t =
@[simp]
theorem set.singleton_sub {α : Type u_2} [has_sub α] {t : set α} {a : α} :
{a} - t =
@[simp]
theorem set.singleton_sub_singleton {α : Type u_2} [has_sub α] {a b : α} :
{a} - {b} = {a - b}
@[simp]
theorem set.singleton_div_singleton {α : Type u_2} [has_div α] {a b : α} :
{a} / {b} = {a / b}
theorem set.div_subset_div {α : Type u_2} [has_div α] {s₁ s₂ t₁ t₂ : set α} :
s₁ t₁s₂ t₂s₁ / s₂ t₁ / t₂
theorem set.sub_subset_sub {α : Type u_2} [has_sub α] {s₁ s₂ t₁ t₂ : set α} :
s₁ t₁s₂ t₂s₁ - s₂ t₁ - t₂
theorem set.div_subset_div_left {α : Type u_2} [has_div α] {s t₁ t₂ : set α} :
t₁ t₂s / t₁ s / t₂
theorem set.sub_subset_sub_left {α : Type u_2} [has_sub α] {s t₁ t₂ : set α} :
t₁ t₂s - t₁ s - t₂
theorem set.sub_subset_sub_right {α : Type u_2} [has_sub α] {s₁ s₂ t : set α} :
s₁ s₂s₁ - t s₂ - t
theorem set.div_subset_div_right {α : Type u_2} [has_div α] {s₁ s₂ t : set α} :
s₁ s₂s₁ / t s₂ / t
theorem set.div_subset_iff {α : Type u_2} [has_div α] {s t u : set α} :
s / t u ∀ (x : α), x s∀ (y : α), y tx / y u
theorem set.sub_subset_iff {α : Type u_2} [has_sub α] {s t u : set α} :
s - t u ∀ (x : α), x s∀ (y : α), y tx - y u
theorem set.union_div {α : Type u_2} [has_div α] {s₁ s₂ t : set α} :
(s₁ s₂) / t = s₁ / t s₂ / t
theorem set.union_sub {α : Type u_2} [has_sub α] {s₁ s₂ t : set α} :
s₁ s₂ - t = s₁ - t (s₂ - t)
theorem set.div_union {α : Type u_2} [has_div α] {s t₁ t₂ : set α} :
s / (t₁ t₂) = s / t₁ s / t₂
theorem set.sub_union {α : Type u_2} [has_sub α] {s t₁ t₂ : set α} :
s - (t₁ t₂) = s - t₁ (s - t₂)
theorem set.inter_div_subset {α : Type u_2} [has_div α] {s₁ s₂ t : set α} :
s₁ s₂ / t s₁ / t (s₂ / t)
theorem set.inter_sub_subset {α : Type u_2} [has_sub α] {s₁ s₂ t : set α} :
s₁ s₂ - t (s₁ - t) (s₂ - t)
theorem set.div_inter_subset {α : Type u_2} [has_div α] {s t₁ t₂ : set α} :
s / (t₁ t₂) s / t₁ (s / t₂)
theorem set.sub_inter_subset {α : Type u_2} [has_sub α] {s t₁ t₂ : set α} :
s - t₁ t₂ (s - t₁) (s - t₂)
theorem set.Union_sub_left_image {α : Type u_2} [has_sub α] {s t : set α} :
(⋃ (a : α) (H : a s), '' t) = s - t
theorem set.Union_div_left_image {α : Type u_2} [has_div α] {s t : set α} :
(⋃ (a : α) (H : a s), '' t) = s / t
theorem set.Union_div_right_image {α : Type u_2} [has_div α] {s t : set α} :
(⋃ (a : α) (H : a t), (λ (_x : α), _x / a) '' s) = s / t
theorem set.Union_sub_right_image {α : Type u_2} [has_sub α] {s t : set α} :
(⋃ (a : α) (H : a t), (λ (_x : α), _x - a) '' s) = s - t
theorem set.Union_div {α : Type u_2} {ι : Sort u_5} [has_div α] (s : ι → set α) (t : set α) :
(⋃ (i : ι), s i) / t = ⋃ (i : ι), s i / t
theorem set.Union_sub {α : Type u_2} {ι : Sort u_5} [has_sub α] (s : ι → set α) (t : set α) :
(⋃ (i : ι), s i) - t = ⋃ (i : ι), s i - t
theorem set.sub_Union {α : Type u_2} {ι : Sort u_5} [has_sub α] (s : set α) (t : ι → set α) :
(s - ⋃ (i : ι), t i) = ⋃ (i : ι), s - t i
theorem set.div_Union {α : Type u_2} {ι : Sort u_5} [has_div α] (s : set α) (t : ι → set α) :
(s / ⋃ (i : ι), t i) = ⋃ (i : ι), s / t i
theorem set.Union₂_sub {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_sub α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋃ (i : ι) (j : κ i), s i j) - t = ⋃ (i : ι) (j : κ i), s i j - t
theorem set.Union₂_div {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_div α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋃ (i : ι) (j : κ i), s i j) / t = ⋃ (i : ι) (j : κ i), s i j / t
theorem set.div_Union₂ {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_div α] (s : set α) (t : Π (i : ι), κ iset α) :
(s / ⋃ (i : ι) (j : κ i), t i j) = ⋃ (i : ι) (j : κ i), s / t i j
theorem set.sub_Union₂ {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_sub α] (s : set α) (t : Π (i : ι), κ iset α) :
(s - ⋃ (i : ι) (j : κ i), t i j) = ⋃ (i : ι) (j : κ i), s - t i j
theorem set.Inter_sub_subset {α : Type u_2} {ι : Sort u_5} [has_sub α] (s : ι → set α) (t : set α) :
(⋂ (i : ι), s i) - t ⋂ (i : ι), s i - t
theorem set.Inter_div_subset {α : Type u_2} {ι : Sort u_5} [has_div α] (s : ι → set α) (t : set α) :
(⋂ (i : ι), s i) / t ⋂ (i : ι), s i / t
theorem set.sub_Inter_subset {α : Type u_2} {ι : Sort u_5} [has_sub α] (s : set α) (t : ι → set α) :
(s - ⋂ (i : ι), t i) ⋂ (i : ι), s - t i
theorem set.div_Inter_subset {α : Type u_2} {ι : Sort u_5} [has_div α] (s : set α) (t : ι → set α) :
(s / ⋂ (i : ι), t i) ⋂ (i : ι), s / t i
theorem set.Inter₂_div_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_div α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋂ (i : ι) (j : κ i), s i j) / t ⋂ (i : ι) (j : κ i), s i j / t
theorem set.Inter₂_sub_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_sub α] (s : Π (i : ι), κ iset α) (t : set α) :
(⋂ (i : ι) (j : κ i), s i j) - t ⋂ (i : ι) (j : κ i), s i j - t
theorem set.div_Inter₂_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_div α] (s : set α) (t : Π (i : ι), κ iset α) :
(s / ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), s / t i j
theorem set.sub_Inter₂_subset {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [has_sub α] (s : set α) (t : Π (i : ι), κ iset α) :
(s - ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), s - t i j
@[protected]
def set.has_nsmul {α : Type u_2} [has_zero α] [has_add α] :
(set α)

Repeated pointwise addition (not the same as pointwise repeated addition!) of a `finset`. See note [pointwise nat action].

Equations
@[protected]
def set.has_npow {α : Type u_2} [has_one α] [has_mul α] :

Repeated pointwise multiplication (not the same as pointwise repeated multiplication!) of a `set`. See note [pointwise nat action].

Equations
@[protected]
def set.has_zsmul {α : Type u_2} [has_zero α] [has_add α] [has_neg α] :
(set α)

Repeated pointwise addition/subtraction (not the same as pointwise repeated addition/subtraction!) of a `set`. See note [pointwise nat action].

Equations
@[protected]
def set.has_zpow {α : Type u_2} [has_one α] [has_mul α] [has_inv α] :

Repeated pointwise multiplication/division (not the same as pointwise repeated multiplication/division!) of a `set`. See note [pointwise nat action].

Equations
@[protected]
def set.add_semigroup {α : Type u_2}  :

`set α` is an `add_semigroup` under pointwise operations if `α` is.

Equations
@[protected]
def set.semigroup {α : Type u_2} [semigroup α] :

`set α` is a `semigroup` under pointwise operations if `α` is.

Equations
@[protected]
def set.add_comm_semigroup {α : Type u_2}  :

`set α` is an `add_comm_semigroup` under pointwise operations if `α` is.

Equations
@[protected]
def set.comm_semigroup {α : Type u_2}  :

`set α` is a `comm_semigroup` under pointwise operations if `α` is.

Equations
@[protected]
def set.add_zero_class {α : Type u_2}  :

`set α` is an `add_zero_class` under pointwise operations if `α` is.

Equations
@[protected]
def set.mul_one_class {α : Type u_2}  :

`set α` is a `mul_one_class` under pointwise operations if `α` is.

Equations
theorem set.subset_mul_left {α : Type u_2} (s : set α) {t : set α} (ht : 1 t) :
s s * t
theorem set.subset_add_left {α : Type u_2} (s : set α) {t : set α} (ht : 0 t) :
s s + t
theorem set.subset_add_right {α : Type u_2} {s : set α} (t : set α) (hs : 0 s) :
t s + t
theorem set.subset_mul_right {α : Type u_2} {s : set α} (t : set α) (hs : 1 s) :
t s * t
def set.singleton_monoid_hom {α : Type u_2}  :
α →* set α

The singleton operation as a `monoid_hom`.

Equations
def set.singleton_add_monoid_hom {α : Type u_2}  :
α →+ set α

The singleton operation as an `add_monoid_hom`.

Equations
@[simp]
theorem set.coe_singleton_monoid_hom {α : Type u_2}  :
@[simp]
theorem set.coe_singleton_add_monoid_hom {α : Type u_2}  :
@[simp]
theorem set.singleton_add_monoid_hom_apply {α : Type u_2} (a : α) :
@[simp]
theorem set.singleton_monoid_hom_apply {α : Type u_2} (a : α) :
@[protected]
def set.monoid {α : Type u_2} [monoid α] :
monoid (set α)

`set α` is a `monoid` under pointwise operations if `α` is.

Equations
@[protected]
def set.add_monoid {α : Type u_2} [add_monoid α] :

`set α` is an `add_monoid` under pointwise operations if `α` is.

Equations
@[protected, instance]
def set.decidable_mem_add {α : Type u_2} [add_monoid α] {s t : set α} [fintype α] [decidable_eq α] [decidable_pred (λ (_x : α), _x s)] [decidable_pred (λ (_x : α), _x t)] :
decidable_pred (λ (_x : α), _x s + t)
Equations
@[protected, instance]
def set.decidable_mem_mul {α : Type u_2} [monoid α] {s t : set α} [fintype α] [decidable_eq α] [decidable_pred (λ (_x : α), _x s)] [decidable_pred (λ (_x : α), _x t)] :
decidable_pred (λ (_x : α), _x s * t)
Equations
@[protected, instance]
def set.decidable_mem_nsmul {α : Type u_2} [add_monoid α] {s : set α} [fintype α] [decidable_eq α] [decidable_pred (λ (_x : α), _x s)] (n : ) :
decidable_pred (λ (_x : α), _x n s)
Equations
• = nat.rec (set.decidable_mem_nsmul._proof_1.mpr (set.decidable_mem_nsmul._proof_2.mpr (λ (a : α), _inst_3 a 0))) (λ (n : ) (ih : decidable_pred (λ (_x : α), _x n s)), let _inst : decidable_pred (λ (_x : α), _x n s) := ih in _.mpr (λ (a : α), n
@[protected, instance]
def set.decidable_mem_pow {α : Type u_2} [monoid α] {s : set α} [fintype α] [decidable_eq α] [decidable_pred (λ (_x : α), _x s)] (n : ) :
decidable_pred (λ (_x : α), _x s ^ n)
Equations
• = nat.rec (set.decidable_mem_pow._proof_1.mpr (set.decidable_mem_pow._proof_2.mpr (λ (a : α), _inst_3 a 1))) (λ (n : ) (ih : decidable_pred (λ (_x : α), _x s ^ n)), let _inst : decidable_pred (λ (_x : α), _x s ^ n) := ih in _.mpr (λ (a : α), n
theorem set.nsmul_mem_nsmul {α : Type u_2} [add_monoid α] {s : set α} {a : α} (ha : a s) (n : ) :
n a n s
theorem set.pow_mem_pow {α : Type u_2} [monoid α] {s : set α} {a : α} (ha : a s) (n : ) :
a ^ n s ^ n
theorem set.nsmul_subset_nsmul {α : Type u_2} [add_monoid α] {s t : set α} (hst : s t) (n : ) :
n s n t
theorem set.pow_subset_pow {α : Type u_2} [monoid α] {s t : set α} (hst : s t) (n : ) :
s ^ n t ^ n
theorem set.pow_subset_pow_of_one_mem {α : Type u_2} [monoid α] {s : set α} {m n : } (hs : 1 s) :
m ns ^ m s ^ n
theorem set.nsmul_subset_nsmul_of_zero_mem {α : Type u_2} [add_monoid α] {s : set α} {m n : } (hs : 0 s) :
m nm s n s
theorem set.mem_prod_list_of_fn {α : Type u_2} [monoid α] {n : } {a : α} {s : fin nset α} :
a (list.of_fn s).prod ∃ (f : Π (i : fin n), (s i)), (list.of_fn (λ (i : fin n), (f i))).prod = a
theorem set.mem_sum_list_of_fn {α : Type u_2} [add_monoid α] {n : } {a : α} {s : fin nset α} :
a (list.of_fn s).sum ∃ (f : Π (i : fin n), (s i)), (list.of_fn (λ (i : fin n), (f i))).sum = a
theorem set.mem_list_sum {α : Type u_2} [add_monoid α] {l : list (set α)} {a : α} :
a l.sum ∃ (l' : list (Σ (s : set α), s)), (list.map (λ (x : Σ (s : set α), s), (x.snd)) l').sum = a = l
theorem set.mem_list_prod {α : Type u_2} [monoid α] {l : list (set α)} {a : α} :
a l.prod ∃ (l' : list (Σ (s : set α), s)), (list.map (λ (x : Σ (s : set α), s), (x.snd)) l').prod = a = l
theorem set.mem_nsmul {α : Type u_2} [add_monoid α] {s : set α} {a : α} {n : } :
a n s ∃ (f : fin ns), (list.of_fn (λ (i : fin n), (f i))).sum = a
theorem set.mem_pow {α : Type u_2} [monoid α] {s : set α} {a : α} {n : } :
a s ^ n ∃ (f : fin ns), (list.of_fn (λ (i : fin n), (f i))).prod = a
@[simp]
theorem set.empty_pow {α : Type u_2} [monoid α] {n : } (hn : n 0) :
@[simp]
theorem set.empty_nsmul {α : Type u_2} [add_monoid α] {n : } (hn : n 0) :
theorem set.mul_univ_of_one_mem {α : Type u_2} [monoid α] {s : set α} (hs : 1 s) :
theorem set.add_univ_of_zero_mem {α : Type u_2} [add_monoid α] {s : set α} (hs : 0 s) :
theorem set.univ_add_of_zero_mem {α : Type u_2} [add_monoid α] {t : set α} (ht : 0 t) :
theorem set.univ_mul_of_one_mem {α : Type u_2} [monoid α] {t : set α} (ht : 1 t) :
@[simp]
theorem set.univ_add_univ {α : Type u_2} [add_monoid α] :
@[simp]
theorem set.univ_mul_univ {α : Type u_2} [monoid α] :
@[simp]
theorem set.nsmul_univ {α : Type u_1} [add_monoid α] {n : } :
n 0
@[simp]
theorem set.univ_pow {α : Type u_2} [monoid α] {n : } :
n 0
@[protected]
theorem is_unit.set {α : Type u_2} [monoid α] {a : α} :
is_unit {a}
@[protected]
theorem is_add_unit.set {α : Type u_2} [add_monoid α] {a : α} :
@[protected]
def set.comm_monoid {α : Type u_2} [comm_monoid α] :

`set α` is a `comm_monoid` under pointwise operations if `α` is.

Equations
@[protected]
def set.add_comm_monoid {α : Type u_2}  :

`set α` is an `add_comm_monoid` under pointwise operations if `α` is.

Equations
@[protected]
theorem set.mul_eq_one_iff {α : Type u_2} {s t : set α} :
s * t = 1 ∃ (a b : α), s = {a} t = {b} a * b = 1
@[protected]
theorem set.add_eq_zero_iff {α : Type u_2} {s t : set α} :
s + t = 0 ∃ (a b : α), s = {a} t = {b} a + b = 0
@[protected]
def set.subtraction_monoid {α : Type u_2}  :

`set α` is a subtraction monoid under pointwise operations if `α` is.

Equations
@[protected]
def set.division_monoid {α : Type u_2}  :

`set α` is a division monoid under pointwise operations if `α` is.

Equations
@[simp]
theorem set.is_unit_iff {α : Type u_2} {s : set α} :
∃ (a : α), s = {a}
@[simp]
theorem set.is_add_unit_iff {α : Type u_2} {s : set α} :
∃ (a : α), s = {a}
@[protected]
def set.division_comm_monoid {α : Type u_2}  :

`set α` is a commutative division monoid under pointwise operations if `α` is.

Equations
@[protected]
def set.subtraction_comm_monoid {α : Type u_2}  :

`set α` is a commutative subtraction monoid under pointwise operations if `α` is.

Equations
@[protected]
def set.has_distrib_neg {α : Type u_2} [has_mul α]  :

`set α` has distributive negation if `α` has.

Equations

Note that `set α` is not a `distrib` because `s * t + s * u` has cross terms that `s * (t + u)` lacks.

theorem set.mul_add_subset {α : Type u_2} [distrib α] (s t u : set α) :
s * (t + u) s * t + s * u
theorem set.add_mul_subset {α : Type u_2} [distrib α] (s t u : set α) :
(s + t) * u s * u + t * u

Note that `set` is not a `mul_zero_class` because `0 * ∅ ≠ 0`.

theorem set.mul_zero_subset {α : Type u_2} (s : set α) :
s * 0 0
theorem set.zero_mul_subset {α : Type u_2} (s : set α) :
0 * s 0
theorem set.nonempty.mul_zero {α : Type u_2} {s : set α} (hs : s.nonempty) :
s * 0 = 0
theorem set.nonempty.zero_mul {α : Type u_2} {s : set α} (hs : s.nonempty) :
0 * s = 0

Note that `set` is not a `group` because `s / s ≠ 1` in general.

@[simp]
theorem set.one_mem_div_iff {α : Type u_2} [group α] {s t : set α} :
1 s / t ¬ t
@[simp]
theorem set.zero_mem_sub_iff {α : Type u_2} [add_group α] {s t : set α} :
0 s - t ¬ t
theorem set.not_zero_mem_sub_iff {α : Type u_2} [add_group α] {s t : set α} :
0 s - t t
theorem set.not_one_mem_div_iff {α : Type u_2} [group α] {s t : set α} :
1 s / t t
theorem disjoint.one_not_mem_div_set {α : Type u_2} [group α] {s t : set α} :
t1 s / t

Alias of the reverse direction of `set.not_one_mem_div_iff`.

theorem disjoint.zero_not_mem_sub_set {α : Type u_2} [add_group α] {s t : set α} :
t0 s - t

Alias of the reverse direction of `set.not_one_mem_div_iff`.

theorem set.nonempty.one_mem_div {α : Type u_2} [group α] {s : set α} (h : s.nonempty) :
1 s / s
theorem set.nonempty.zero_mem_sub {α : Type u_2} [add_group α] {s : set α} (h : s.nonempty) :
0 s - s
theorem set.is_unit_singleton {α : Type u_2} [group α] (a : α) :
theorem set.is_add_unit_singleton {α : Type u_2} [add_group α] (a : α) :
@[simp]
theorem set.is_add_unit_iff_singleton {α : Type u_2} [add_group α] {s : set α} :
∃ (a : α), s = {a}
@[simp]
theorem set.is_unit_iff_singleton {α : Type u_2} [group α] {s : set α} :
∃ (a : α), s = {a}
@[simp]
theorem set.image_add_left {α : Type u_2} [add_group α] {t : set α} {a : α} :
@[simp]
theorem set.image_mul_left {α : Type u_2} [group α] {t : set α} {a : α} :
=
@[simp]
theorem set.image_mul_right {α : Type u_2} [group α] {t : set α} {b : α} :
(λ (_x : α), _x * b) '' t = (λ (_x : α), _x * b⁻¹) ⁻¹' t
@[simp]
theorem set.image_add_right {α : Type u_2} [add_group α] {t : set α} {b : α} :
(λ (_x : α), _x + b) '' t = (λ (_x : α), _x + -b) ⁻¹' t
theorem set.image_mul_left' {α : Type u_2} [group α] {t : set α} {a : α} :
(λ (b : α), a⁻¹ * b) '' t = (λ (b : α), a * b) ⁻¹' t
theorem set.image_add_left' {α : Type u_2} [add_group α] {t : set α} {a : α} :
(λ (b : α), -a + b) '' t = (λ (b : α), a + b) ⁻¹' t
theorem set.image_mul_right' {α : Type u_2} [group α] {t : set α} {b : α} :
(λ (_x : α), _x * b⁻¹) '' t = (λ (_x : α), _x * b) ⁻¹' t
theorem set.image_add_right' {α : Type u_2} [add_group α] {t : set α} {b : α} :
(λ (_x : α), _x + -b) '' t = (λ (_x : α), _x + b) ⁻¹' t
@[simp]
theorem set.preimage_mul_left_singleton {α : Type u_2} [group α] {a b : α} :
⁻¹' {b} = {a⁻¹ * b}
@[simp]
theorem set.preimage_add_left_singleton {α : Type u_2} [add_group α] {a b : α} :
⁻¹' {b} = {-a + b}
@[simp]
theorem set.preimage_mul_right_singleton {α : Type u_2} [group α] {a b : α} :
(λ (_x : α), _x * a) ⁻¹' {b} = {b * a⁻¹}
@[simp]
theorem set.preimage_add_right_singleton {α : Type u_2} [add_group α] {a b : α} :
(λ (_x : α), _x + a) ⁻¹' {b} = {b + -a}
@[simp]
theorem set.preimage_mul_left_one {α : Type u_2} [group α] {a : α} :
= {a⁻¹}
@[simp]
theorem set.preimage_add_left_zero {α : Type u_2} [add_group α] {a : α} :
= {-a}
@[simp]
theorem set.preimage_add_right_zero {α : Type u_2} [add_group α] {b : α} :
(λ (_x : α), _x + b) ⁻¹' 0 = {-b}
@[simp]
theorem set.preimage_mul_right_one {α : Type u_2} [group α] {b : α} :
(λ (_x : α), _x * b) ⁻¹' 1 = {b⁻¹}
theorem set.preimage_add_left_zero' {α : Type u_2} [add_group α] {a : α} :
(λ (b : α), -a + b) ⁻¹' 0 = {a}
theorem set.preimage_mul_left_one' {α : Type u_2} [group α] {a : α} :
(λ (b : α), a⁻¹ * b) ⁻¹' 1 = {a}
theorem set.preimage_add_right_zero' {α : Type u_2} [add_group α] {b : α} :
(λ (_x : α), _x + -b) ⁻¹' 0 = {b}
theorem set.preimage_mul_right_one' {α : Type u_2} [group α] {b : α} :
(λ (_x : α), _x * b⁻¹) ⁻¹' 1 = {b}
@[simp]
theorem set.mul_univ {α : Type u_2} [group α] {s : set α} (hs : s.nonempty) :
@[simp]
theorem set.add_univ {α : Type u_2} [add_group α] {s : set α} (hs : s.nonempty) :
@[simp]
theorem set.univ_mul {α : Type u_2} [group α] {t : set α} (ht : t.nonempty) :
@[simp]
theorem set.univ_add {α : Type u_2} [add_group α] {t : set α} (ht : t.nonempty) :
theorem set.div_zero_subset {α : Type u_2} (s : set α) :
s / 0 0
theorem set.zero_div_subset {α : Type u_2} (s : set α) :
0 / s 0
theorem set.nonempty.div_zero {α : Type u_2} {s : set α} (hs : s.nonempty) :
s / 0 = 0
theorem set.nonempty.zero_div {α : Type u_2} {s : set α} (hs : s.nonempty) :
0 / s = 0
theorem set.image_add {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_add α] [has_add β] [ β] (m : F) {s t : set α} :
m '' (s + t) = m '' s + m '' t
theorem set.image_mul {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_mul α] [has_mul β] [ β] (m : F) {s t : set α} :
m '' (s * t) = m '' s * m '' t
theorem set.preimage_mul_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_mul α] [has_mul β] [ β] (m : F) {s t : set β} :
theorem set.preimage_add_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_add α] [has_add β] [ β] (m : F) {s t : set β} :
theorem set.image_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [ β] (m : F) {s t : set α} :
m '' (s / t) = m '' s / m '' t
theorem set.image_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [ β] (m : F) {s t : set α} :
m '' (s - t) = m '' s - m '' t
theorem set.preimage_div_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [ β] (m : F) {s t : set β} :
theorem set.preimage_sub_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [ β] (m : F) {s t : set β} :
theorem set.bdd_above_add {α : Type u_2} {A B : set α} :
bdd_above (A + B)
theorem set.bdd_above_mul {α : Type u_2} {A B : set α} :
bdd_above (A * B)
theorem set.mem_finset_sum {α : Type u_2} {ι : Type u_5} (t : finset ι) (f : ι → set α) (a : α) :
a t.sum (λ (i : ι), f i) ∃ (g : ι → α) (hg : ∀ {i : ι}, i tg i f i), t.sum (λ (i : ι), g i) = a

The n-ary version of `set.mem_add`.

theorem set.mem_finset_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : finset ι) (f : ι → set α) (a : α) :
a t.prod (λ (i : ι), f i) ∃ (g : ι → α) (hg : ∀ {i : ι}, i tg i f i), t.prod (λ (i : ι), g i) = a

The n-ary version of `set.mem_mul`.

theorem set.mem_fintype_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] [fintype ι] (f : ι → set α) (a : α) :
a finset.univ.prod (λ (i : ι), f i) ∃ (g : ι → α) (hg : ∀ (i : ι), g i f i), finset.univ.prod (λ (i : ι), g i) = a

A version of `set.mem_finset_prod` with a simpler RHS for products over a fintype.

theorem set.mem_fintype_sum {α : Type u_2} {ι : Type u_5} [fintype ι] (f : ι → set α) (a : α) :
a finset.univ.sum (λ (i : ι), f i) ∃ (g : ι → α) (hg : ∀ (i : ι), g i f i), finset.univ.sum (λ (i : ι), g i) = a

A version of `set.mem_finset_sum` with a simpler RHS for sums over a fintype.

theorem set.list_sum_mem_list_sum {α : Type u_2} {ι : Type u_5} (t : list ι) (f : ι → set α) (g : ι → α) (hg : ∀ (i : ι), i tg i f i) :

An n-ary version of `set.add_mem_add`.

theorem set.list_prod_mem_list_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : list ι) (f : ι → set α) (g : ι → α) (hg : ∀ (i : ι), i tg i f i) :

An n-ary version of `set.mul_mem_mul`.

theorem set.list_sum_subset_list_sum {α : Type u_2} {ι : Type u_5} (t : list ι) (f₁ f₂ : ι → set α) (hf : ∀ (i : ι), i tf₁ i f₂ i) :
(list.map f₁ t).sum (list.map f₂ t).sum

An n-ary version of `set.add_subset_add`.

theorem set.list_prod_subset_list_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : list ι) (f₁ f₂ : ι → set α) (hf : ∀ (i : ι), i tf₁ i f₂ i) :
(list.map f₁ t).prod (list.map f₂ t).prod

An n-ary version of `set.mul_subset_mul`.

theorem set.list_sum_singleton {M : Type u_1} (s : list M) :
(list.map (λ (i : M), {i}) s).sum = {s.sum}
theorem set.list_prod_singleton {M : Type u_1} [comm_monoid M] (s : list M) :
(list.map (λ (i : M), {i}) s).prod = {s.prod}
theorem set.multiset_sum_mem_multiset_sum {α : Type u_2} {ι : Type u_5} (t : multiset ι) (f : ι → set α) (g : ι → α) (hg : ∀ (i : ι), i tg i f i) :
t).sum t).sum

An n-ary version of `set.add_mem_add`.

theorem set.multiset_prod_mem_multiset_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : multiset ι) (f : ι → set α) (g : ι → α) (hg : ∀ (i : ι), i tg i f i) :
t).prod t).prod

An n-ary version of `set.mul_mem_mul`.

theorem set.multiset_prod_subset_multiset_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : multiset ι) (f₁ f₂ : ι → set α) (hf : ∀ (i : ι), i tf₁ i f₂ i) :

An n-ary version of `set.mul_subset_mul`.

theorem set.multiset_sum_subset_multiset_sum {α : Type u_2} {ι : Type u_5} (t : multiset ι) (f₁ f₂ : ι → set α) (hf : ∀ (i : ι), i tf₁ i f₂ i) :
(multiset.map f₁ t).sum (multiset.map f₂ t).sum

An n-ary version of `set.add_subset_add`.

theorem set.multiset_sum_singleton {M : Type u_1} (s : multiset M) :
(multiset.map (λ (i : M), {i}) s).sum = {s.sum}
theorem set.multiset_prod_singleton {M : Type u_1} [comm_monoid M] (s : multiset M) :
(multiset.map (λ (i : M), {i}) s).prod = {s.prod}
theorem set.finset_prod_mem_finset_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : finset ι) (f : ι → set α) (g : ι → α) (hg : ∀ (i : ι), i tg i f i) :
t.prod (λ (i : ι), g i) t.prod (λ (i : ι), f i)

An n-ary version of `set.mul_mem_mul`.

theorem set.finset_sum_mem_finset_sum {α : Type u_2} {ι : Type u_5} (t : finset ι) (f : ι → set α) (g : ι → α) (hg : ∀ (i : ι), i tg i f i) :
t.sum (λ (i : ι), g i) t.sum (λ (i : ι), f i)

An n-ary version of `set.add_mem_add`.

theorem set.finset_prod_subset_finset_prod {α : Type u_2} {ι : Type u_5} [comm_monoid α] (t : finset ι) (f₁ f₂ : ι → set α) (hf : ∀ (i : ι), i tf₁ i f₂ i) :
t.prod (λ (i : ι), f₁ i) t.prod (λ (i : ι), f₂ i)

An n-ary version of `set.mul_subset_mul`.

theorem set.finset_sum_subset_finset_sum {α : Type u_2} {ι : Type u_5} (t : finset ι) (f₁ f₂ : ι → set α) (hf : ∀ (i : ι), i tf₁ i f₂ i) :
t.sum (λ (i : ι), f₁ i) t.sum (λ (i : ι), f₂ i)

An n-ary version of `set.add_subset_add`.

theorem set.finset_prod_singleton {M : Type u_1} {ι : Type u_2} [comm_monoid M] (s : finset ι) (I : ι → M) :
s.prod (λ (i : ι), {I i}) = {s.prod (λ (i : ι), I i)}
theorem set.finset_sum_singleton {M : Type u_1} {ι : Type u_2} (s : finset ι) (I : ι → M) :
s.sum (λ (i : ι), {I i}) = {s.sum (λ (i : ι), I i)}

TODO: define `decidable_mem_finset_prod` and `decidable_mem_finset_sum`.

### Translation/scaling of sets #

@[protected]
def set.has_smul_set {α : Type u_2} {β : Type u_3} [ β] :
(set β)

The dilation of set `x • s` is defined as `{x • y | y ∈ s}` in locale `pointwise`.

Equations
@[protected]
def set.has_vadd_set {α : Type u_2} {β : Type u_3} [ β] :
(set β)

The translation of set `x +ᵥ s` is defined as `{x +ᵥ y | y ∈ s}` in locale `pointwise`.

Equations
@[protected]
def set.has_smul {α : Type u_2} {β : Type u_3} [ β] :
has_smul (set α) (set β)

The pointwise scalar multiplication of sets `s • t` is defined as `{x • y | x ∈ s, y ∈ t}` in locale `pointwise`.

Equations
@[protected]
def set.has_vadd {α : Type u_2} {β : Type u_3} [ β] :
has_vadd (set α) (set β)

The pointwise scalar addition of sets `s +ᵥ t` is defined as `{x +ᵥ y | x ∈ s, y ∈ t}` in locale `pointwise`.

Equations
@[simp]
theorem set.image2_smul {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
= s t
@[simp]
theorem set.image2_vadd {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
= s +ᵥ t
theorem set.image_smul_prod {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(λ (x : α × β), x.fst x.snd) '' s ×ˢ t = s t
theorem set.mem_vadd {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} {b : β} :
b s +ᵥ t ∃ (x : α) (y : β), x s y t x +ᵥ y = b
theorem set.mem_smul {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} {b : β} :
b s t ∃ (x : α) (y : β), x s y t x y = b
theorem set.vadd_mem_vadd {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} {a : α} {b : β} :
a sb ta +ᵥ b s +ᵥ t
theorem set.smul_mem_smul {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} {a : α} {b : β} :
a sb ta b s t
@[simp]
theorem set.empty_vadd {α : Type u_2} {β : Type u_3} [ β] {t : set β} :
@[simp]
theorem set.empty_smul {α : Type u_2} {β : Type u_3} [ β] {t : set β} :
@[simp]
theorem set.smul_empty {α : Type u_2} {β : Type u_3} [ β] {s : set α} :
@[simp]
theorem set.vadd_empty {α : Type u_2} {β : Type u_3} [ β] {s : set α} :
@[simp]
theorem set.vadd_eq_empty {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
s +ᵥ t = s = t =
@[simp]
theorem set.smul_eq_empty {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
s t = s = t =
@[simp]
theorem set.smul_nonempty {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
@[simp]
theorem set.vadd_nonempty {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
theorem set.nonempty.smul {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
s.nonemptyt.nonempty(s t).nonempty
theorem set.nonempty.vadd {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
s.nonemptyt.nonempty(s +ᵥ t).nonempty
theorem set.nonempty.of_smul_left {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(s t).nonempty → s.nonempty
theorem set.nonempty.of_vadd_left {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(s +ᵥ t).nonempty → s.nonempty
theorem set.nonempty.of_vadd_right {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(s +ᵥ t).nonempty → t.nonempty
theorem set.nonempty.of_smul_right {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(s t).nonempty → t.nonempty
@[simp]
theorem set.vadd_singleton {α : Type u_2} {β : Type u_3} [ β] {s : set α} {b : β} :
s +ᵥ {b} = (λ (_x : α), _x +ᵥ b) '' s
@[simp]
theorem set.smul_singleton {α : Type u_2} {β : Type u_3} [ β] {s : set α} {b : β} :
s {b} = (λ (_x : α), _x b) '' s
@[simp]
theorem set.singleton_smul {α : Type u_2} {β : Type u_3} [ β] {t : set β} {a : α} :
{a} t = a t
@[simp]
theorem set.singleton_vadd {α : Type u_2} {β : Type u_3} [ β] {t : set β} {a : α} :
{a} +ᵥ t = a +ᵥ t
@[simp]
theorem set.singleton_vadd_singleton {α : Type u_2} {β : Type u_3} [ β] {a : α} {b : β} :
{a} +ᵥ {b} = {a +ᵥ b}
@[simp]
theorem set.singleton_smul_singleton {α : Type u_2} {β : Type u_3} [ β] {a : α} {b : β} :
{a} {b} = {a b}
theorem set.smul_subset_smul {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t₁ t₂ : set β} :
s₁ s₂t₁ t₂s₁ t₁ s₂ t₂
theorem set.vadd_subset_vadd {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t₁ t₂ : set β} :
s₁ s₂t₁ t₂s₁ +ᵥ t₁ s₂ +ᵥ t₂
theorem set.vadd_subset_vadd_left {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t₁ t₂ : set β} :
t₁ t₂s +ᵥ t₁ s +ᵥ t₂
theorem set.smul_subset_smul_left {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t₁ t₂ : set β} :
t₁ t₂s t₁ s t₂
theorem set.vadd_subset_vadd_right {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t : set β} :
s₁ s₂s₁ +ᵥ t s₂ +ᵥ t
theorem set.smul_subset_smul_right {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t : set β} :
s₁ s₂s₁ t s₂ t
theorem set.vadd_subset_iff {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t u : set β} :
s +ᵥ t u ∀ (a : α), a s∀ (b : β), b ta +ᵥ b u
theorem set.smul_subset_iff {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t u : set β} :
s t u ∀ (a : α), a s∀ (b : β), b ta b u
theorem set.union_smul {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t : set β} :
(s₁ s₂) t = s₁ t s₂ t
theorem set.union_vadd {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t : set β} :
s₁ s₂ +ᵥ t = s₁ +ᵥ t (s₂ +ᵥ t)
theorem set.smul_union {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t₁ t₂ : set β} :
s (t₁ t₂) = s t₁ s t₂
theorem set.vadd_union {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t₁ t₂ : set β} :
s +ᵥ (t₁ t₂) = s +ᵥ t₁ (s +ᵥ t₂)
theorem set.inter_vadd_subset {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t : set β} :
s₁ s₂ +ᵥ t (s₁ +ᵥ t) (s₂ +ᵥ t)
theorem set.inter_smul_subset {α : Type u_2} {β : Type u_3} [ β] {s₁ s₂ : set α} {t : set β} :
(s₁ s₂) t s₁ t s₂ t
theorem set.vadd_inter_subset {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t₁ t₂ : set β} :
s +ᵥ t₁ t₂ (s +ᵥ t₁) (s +ᵥ t₂)
theorem set.smul_inter_subset {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t₁ t₂ : set β} :
s (t₁ t₂) s t₁ s t₂
theorem set.Union_smul_left_image {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(⋃ (a : α) (H : a s), a t) = s t
theorem set.Union_vadd_left_image {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(⋃ (a : α) (H : a s), a +ᵥ t) = s +ᵥ t
theorem set.Union_smul_right_image {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(⋃ (a : β) (H : a t), (λ (_x : α), _x a) '' s) = s t
theorem set.Union_vadd_right_image {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
(⋃ (a : β) (H : a t), (λ (_x : α), _x +ᵥ a) '' s) = s +ᵥ t
theorem set.Union_smul {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : ι → set α) (t : set β) :
(⋃ (i : ι), s i) t = ⋃ (i : ι), s i t
theorem set.Union_vadd {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : ι → set α) (t : set β) :
(⋃ (i : ι), s i) +ᵥ t = ⋃ (i : ι), s i +ᵥ t
theorem set.smul_Union {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : set α) (t : ι → set β) :
(s ⋃ (i : ι), t i) = ⋃ (i : ι), s t i
theorem set.vadd_Union {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : set α) (t : ι → set β) :
(s +ᵥ ⋃ (i : ι), t i) = ⋃ (i : ι), s +ᵥ t i
theorem set.Union₂_smul {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : Π (i : ι), κ iset α) (t : set β) :
(⋃ (i : ι) (j : κ i), s i j) t = ⋃ (i : ι) (j : κ i), s i j t
theorem set.Union₂_vadd {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : Π (i : ι), κ iset α) (t : set β) :
(⋃ (i : ι) (j : κ i), s i j) +ᵥ t = ⋃ (i : ι) (j : κ i), s i j +ᵥ t
theorem set.vadd_Union₂ {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : set α) (t : Π (i : ι), κ iset β) :
(s +ᵥ ⋃ (i : ι) (j : κ i), t i j) = ⋃ (i : ι) (j : κ i), s +ᵥ t i j
theorem set.smul_Union₂ {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : set α) (t : Π (i : ι), κ iset β) :
(s ⋃ (i : ι) (j : κ i), t i j) = ⋃ (i : ι) (j : κ i), s t i j
theorem set.Inter_vadd_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : ι → set α) (t : set β) :
(⋂ (i : ι), s i) +ᵥ t ⋂ (i : ι), s i +ᵥ t
theorem set.Inter_smul_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : ι → set α) (t : set β) :
(⋂ (i : ι), s i) t ⋂ (i : ι), s i t
theorem set.vadd_Inter_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : set α) (t : ι → set β) :
(s +ᵥ ⋂ (i : ι), t i) ⋂ (i : ι), s +ᵥ t i
theorem set.smul_Inter_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (s : set α) (t : ι → set β) :
(s ⋂ (i : ι), t i) ⋂ (i : ι), s t i
theorem set.Inter₂_vadd_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : Π (i : ι), κ iset α) (t : set β) :
(⋂ (i : ι) (j : κ i), s i j) +ᵥ t ⋂ (i : ι) (j : κ i), s i j +ᵥ t
theorem set.Inter₂_smul_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : Π (i : ι), κ iset α) (t : set β) :
(⋂ (i : ι) (j : κ i), s i j) t ⋂ (i : ι) (j : κ i), s i j t
theorem set.vadd_Inter₂_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : set α) (t : Π (i : ι), κ iset β) :
(s +ᵥ ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), s +ᵥ t i j
theorem set.smul_Inter₂_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (s : set α) (t : Π (i : ι), κ iset β) :
(s ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), s t i j
theorem set.finite.smul {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
s.finitet.finite(s t).finite
theorem set.finite.vadd {α : Type u_2} {β : Type u_3} [ β] {s : set α} {t : set β} :
s.finitet.finite(s +ᵥ t).finite
@[simp]
theorem set.bUnion_smul_set {α : Type u_2} {β : Type u_3} [ β] (s : set α) (t : set β) :
(⋃ (a : α) (H : a s), a t) = s t
@[simp]
theorem set.bUnion_vadd_set {α : Type u_2} {β : Type u_3} [ β] (s : set α) (t : set β) :
(⋃ (a : α) (H : a s), a +ᵥ t) = s +ᵥ t
@[simp]
theorem set.image_vadd {α : Type u_2} {β : Type u_3} [ β] {t : set β} {a : α} :
(λ (x : β), a +ᵥ x) '' t = a +ᵥ t
@[simp]
theorem set.image_smul {α : Type u_2} {β : Type u_3} [ β] {t : set β} {a : α} :
(λ (x : β), a x) '' t = a t
theorem set.mem_vadd_set {α : Type u_2} {β : Type u_3} [ β] {t : set β} {a : α} {x : β} :
x a +ᵥ t ∃ (y : β), y t a +ᵥ y = x
theorem set.mem_smul_set {α : Type u_2} {β : Type u_3} [ β] {t : set β} {a : α} {x : β} :
x a t ∃ (y : β), y t a y = x
theorem set.smul_mem_smul_set {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} {b : β} :
b sa b a s
theorem set.vadd_mem_vadd_set {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} {b : β} :
b sa +ᵥ b a +ᵥ s
@[simp]
theorem set.smul_set_empty {α : Type u_2} {β : Type u_3} [ β] {a : α} :
@[simp]
theorem set.vadd_set_empty {α : Type u_2} {β : Type u_3} [ β] {a : α} :
@[simp]
theorem set.smul_set_eq_empty {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} :
a s = s =
@[simp]
theorem set.vadd_set_eq_empty {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} :
a +ᵥ s = s =
@[simp]
theorem set.smul_set_nonempty {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} :
@[simp]
theorem set.vadd_set_nonempty {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} :
@[simp]
theorem set.vadd_set_singleton {α : Type u_2} {β : Type u_3} [ β] {a : α} {b : β} :
a +ᵥ {b} = {a +ᵥ b}
@[simp]
theorem set.smul_set_singleton {α : Type u_2} {β : Type u_3} [ β] {a : α} {b : β} :
a {b} = {a b}
theorem set.vadd_set_mono {α : Type u_2} {β : Type u_3} [ β] {s t : set β} {a : α} :
s ta +ᵥ s a +ᵥ t
theorem set.smul_set_mono {α : Type u_2} {β : Type u_3} [ β] {s t : set β} {a : α} :
s ta s a t
theorem set.vadd_set_subset_iff {α : Type u_2} {β : Type u_3} [ β] {s t : set β} {a : α} :
a +ᵥ s t ∀ ⦃b : β⦄, b sa +ᵥ b t
theorem set.smul_set_subset_iff {α : Type u_2} {β : Type u_3} [ β] {s t : set β} {a : α} :
a s t ∀ ⦃b : β⦄, b sa b t
theorem set.vadd_set_union {α : Type u_2} {β : Type u_3} [ β] {t₁ t₂ : set β} {a : α} :
a +ᵥ (t₁ t₂) = a +ᵥ t₁ (a +ᵥ t₂)
theorem set.smul_set_union {α : Type u_2} {β : Type u_3} [ β] {t₁ t₂ : set β} {a : α} :
a (t₁ t₂) = a t₁ a t₂
theorem set.smul_set_inter_subset {α : Type u_2} {β : Type u_3} [ β] {t₁ t₂ : set β} {a : α} :
a (t₁ t₂) a t₁ a t₂
theorem set.vadd_set_inter_subset {α : Type u_2} {β : Type u_3} [ β] {t₁ t₂ : set β} {a : α} :
a +ᵥ t₁ t₂ (a +ᵥ t₁) (a +ᵥ t₂)
theorem set.vadd_set_Union {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (a : α) (s : ι → set β) :
(a +ᵥ ⋃ (i : ι), s i) = ⋃ (i : ι), a +ᵥ s i
theorem set.smul_set_Union {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (a : α) (s : ι → set β) :
(a ⋃ (i : ι), s i) = ⋃ (i : ι), a s i
theorem set.smul_set_Union₂ {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (a : α) (s : Π (i : ι), κ iset β) :
(a ⋃ (i : ι) (j : κ i), s i j) = ⋃ (i : ι) (j : κ i), a s i j
theorem set.vadd_set_Union₂ {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (a : α) (s : Π (i : ι), κ iset β) :
(a +ᵥ ⋃ (i : ι) (j : κ i), s i j) = ⋃ (i : ι) (j : κ i), a +ᵥ s i j
theorem set.vadd_set_Inter_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (a : α) (t : ι → set β) :
(a +ᵥ ⋂ (i : ι), t i) ⋂ (i : ι), a +ᵥ t i
theorem set.smul_set_Inter_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [ β] (a : α) (t : ι → set β) :
(a ⋂ (i : ι), t i) ⋂ (i : ι), a t i
theorem set.vadd_set_Inter₂_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (a : α) (t : Π (i : ι), κ iset β) :
(a +ᵥ ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), a +ᵥ t i j
theorem set.smul_set_Inter₂_subset {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [ β] (a : α) (t : Π (i : ι), κ iset β) :
(a ⋂ (i : ι) (j : κ i), t i j) ⋂ (i : ι) (j : κ i), a t i j
theorem set.nonempty.vadd_set {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} :
s.nonempty(a +ᵥ s).nonempty
theorem set.nonempty.smul_set {α : Type u_2} {β : Type u_3} [ β] {s : set β} {a : α} :
s.nonempty(a s).nonempty
theorem