mathlib documentation

data.set.opposite

The opposite of a set #

The opposite of a set s is simply the set obtained by taking the opposite of each member of s.

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@[protected]
def set.op {α : Type u_1} (s : set α) :
set αᵒᵖ

The opposite of a set s is the set obtained by taking the opposite of each member of s.

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@[protected]
def set.unop {α : Type u_1} (s : set αᵒᵖ) :
set α

The unop of a set s is the set obtained by taking the unop of each member of s.

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@[simp]
theorem set.mem_op {α : Type u_1} {s : set α} {a : αᵒᵖ} :
a ∈ s.op ↔ opposite.unop a ∈ s
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@[simp]
theorem set.op_mem_op {α : Type u_1} {s : set α} {a : α} :
opposite.op a ∈ s.op ↔ a ∈ s
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@[simp]
theorem set.mem_unop {α : Type u_1} {s : set αᵒᵖ} {a : α} :
a ∈ s.unop ↔ opposite.op a ∈ s
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@[simp]
theorem set.unop_mem_unop {α : Type u_1} {s : set αᵒᵖ} {a : αᵒᵖ} :
opposite.unop a ∈ s.unop ↔ a ∈ s
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@[simp]
theorem set.op_unop {α : Type u_1} (s : set α) :
s.op.unop = s
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@[simp]
theorem set.unop_op {α : Type u_1} (s : set αᵒᵖ) :
s.unop.op = s
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@[simp]
theorem set.op_equiv_self_apply_coe {α : Type u_1} (s : set α) (x : ↥(s.op)) :
↑(⇑(s.op_equiv_self) x) = opposite.unop ↑x
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@[simp]
theorem set.op_equiv_self_symm_apply_coe {α : Type u_1} (s : set α) (x : ↥s) :
↑(⇑(s.op_equiv_self.symm) x) = opposite.op ↑x
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def set.op_equiv_self {α : Type u_1} (s : set α) :
↥(s.op) ≃ ↥s

The members of the opposite of a set are in bijection with the members of the set itself.

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@[simp]
theorem set.op_equiv_apply {α : Type u_1} (s : set α) :
⇑set.op_equiv s = s.op
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def set.op_equiv {α : Type u_1} :
set α ≃ set αᵒᵖ

Taking opposites as an equivalence of powersets.

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@[simp]
theorem set.op_equiv_symm_apply {α : Type u_1} (s : set αᵒᵖ) :
⇑(set.op_equiv.symm) s = s.unop
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@[simp]
theorem set.singleton_op {α : Type u_1} (x : α) :
{x}.op = {opposite.op x}
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@[simp]
theorem set.singleton_unop {α : Type u_1} (x : αᵒᵖ) :
{x}.unop = {opposite.unop x}
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@[simp]
theorem set.singleton_op_unop {α : Type u_1} (x : α) :
{opposite.op x}.unop = {x}
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@[simp]
theorem set.singleton_unop_op {α : Type u_1} (x : αᵒᵖ) :
{opposite.unop x}.op = {x}