Documentation

Mathlib.Algebra.Order.Interval.Finset.Basic

Algebraic properties of finset intervals #

This file provides results about the interaction of algebra with Finset.Ixx.

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@[simp]
theorem Finset.map_add_left_Icc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addLeftEmbedding c) (Icc a b) = Icc (c + a) (c + b)
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@[simp]
theorem Finset.map_add_right_Icc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addRightEmbedding c) (Icc a b) = Icc (a + c) (b + c)
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@[simp]
theorem Finset.map_add_left_Ico {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addLeftEmbedding c) (Ico a b) = Ico (c + a) (c + b)
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@[simp]
theorem Finset.map_add_right_Ico {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addRightEmbedding c) (Ico a b) = Ico (a + c) (b + c)
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@[simp]
theorem Finset.map_add_left_Ioc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addLeftEmbedding c) (Ioc a b) = Ioc (c + a) (c + b)
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@[simp]
theorem Finset.map_add_right_Ioc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addRightEmbedding c) (Ioc a b) = Ioc (a + c) (b + c)
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@[simp]
theorem Finset.map_add_left_Ioo {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addLeftEmbedding c) (Ioo a b) = Ioo (c + a) (c + b)
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@[simp]
theorem Finset.map_add_right_Ioo {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α) :
map (addRightEmbedding c) (Ioo a b) = Ioo (a + c) (b + c)
source
@[simp]
theorem Finset.image_add_left_Icc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => c + x) (Icc a b) = Icc (c + a) (c + b)
source
@[simp]
theorem Finset.image_add_left_Ico {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => c + x) (Ico a b) = Ico (c + a) (c + b)
source
@[simp]
theorem Finset.image_add_left_Ioc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => c + x) (Ioc a b) = Ioc (c + a) (c + b)
source
@[simp]
theorem Finset.image_add_left_Ioo {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => c + x) (Ioo a b) = Ioo (c + a) (c + b)
source
@[simp]
theorem Finset.image_add_right_Icc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => x + c) (Icc a b) = Icc (a + c) (b + c)
source
@[simp]
theorem Finset.image_add_right_Ico {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => x + c) (Ico a b) = Ico (a + c) (b + c)
source
@[simp]
theorem Finset.image_add_right_Ioc {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => x + c) (Ioc a b) = Ioc (a + c) (b + c)
source
@[simp]
theorem Finset.image_add_right_Ioo {α : Type u_1} [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α) :
image (fun (x : α) => x + c) (Ioo a b) = Ioo (a + c) (b + c)