Documentation

Mathlib.Algebra.Order.Pointwise

Pointwise operations on ordered algebraic objects #

This file contains lemmas about the effect of pointwise operations on sets with an order structure.

TODO #

sSup (s • t) = sSup s • sSup t and sInf (s • t) = sInf s • sInf t hold as well but CovariantClass is currently not polymorphic enough to state it.

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@[simp]
theorem sSup_zero {α : Type u_1} [CompleteLattice α] [Zero α] :
sSup 0 = 0
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@[simp]
theorem sSup_one {α : Type u_1} [CompleteLattice α] [One α] :
sSup 1 = 1
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@[simp]
theorem sInf_zero {α : Type u_1} [CompleteLattice α] [Zero α] :
sInf 0 = 0
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@[simp]
theorem sInf_one {α : Type u_1} [CompleteLattice α] [One α] :
sInf 1 = 1
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theorem sSup_neg {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) :
sSup (-s) = -sInf s
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theorem sSup_inv {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) :
sSup s⁻¹ = (sInf s)⁻¹
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theorem sInf_neg {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) :
sInf (-s) = -sSup s
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theorem sInf_inv {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) :
sInf s⁻¹ = (sSup s)⁻¹
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theorem sSup_add {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sSup (s + t) = sSup s + sSup t
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theorem sSup_mul {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sSup (s * t) = sSup s * sSup t
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theorem sInf_add {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sInf (s + t) = sInf s + sInf t
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theorem sInf_mul {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sInf (s * t) = sInf s * sInf t
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theorem sSup_sub {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sSup (s - t) = sSup s - sInf t
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theorem sSup_div {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sSup (s / t) = sSup s / sInf t
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theorem sInf_sub {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sInf (s - t) = sInf s - sSup t
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theorem sInf_div {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Set α) (t : Set α) :
sInf (s / t) = sInf s / sSup t
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@[simp]
theorem csSup_zero {α : Type u_1} [ConditionallyCompleteLattice α] [Zero α] :
sSup 0 = 0
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@[simp]
theorem csSup_one {α : Type u_1} [ConditionallyCompleteLattice α] [One α] :
sSup 1 = 1
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@[simp]
theorem csInf_zero {α : Type u_1} [ConditionallyCompleteLattice α] [Zero α] :
sInf 0 = 0
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@[simp]
theorem csInf_one {α : Type u_1} [ConditionallyCompleteLattice α] [One α] :
sInf 1 = 1
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theorem csSup_neg {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddBelow s) :
sSup (-s) = -sInf s
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theorem csSup_inv {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddBelow s) :
sSup s⁻¹ = (sInf s)⁻¹
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theorem csInf_neg {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddAbove s) :
sInf (-s) = -sSup s
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theorem csInf_inv {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddAbove s) :
sInf s⁻¹ = (sSup s)⁻¹
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theorem csSup_add {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddAbove s) (ht₀ : Set.Nonempty t) (ht₁ : BddAbove t) :
sSup (s + t) = sSup s + sSup t
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theorem csSup_mul {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddAbove s) (ht₀ : Set.Nonempty t) (ht₁ : BddAbove t) :
sSup (s * t) = sSup s * sSup t
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theorem csInf_add {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddBelow s) (ht₀ : Set.Nonempty t) (ht₁ : BddBelow t) :
sInf (s + t) = sInf s + sInf t
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theorem csInf_mul {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddBelow s) (ht₀ : Set.Nonempty t) (ht₁ : BddBelow t) :
sInf (s * t) = sInf s * sInf t
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theorem csSup_sub {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddAbove s) (ht₀ : Set.Nonempty t) (ht₁ : BddBelow t) :
sSup (s - t) = sSup s - sInf t
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theorem csSup_div {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddAbove s) (ht₀ : Set.Nonempty t) (ht₁ : BddBelow t) :
sSup (s / t) = sSup s / sInf t
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theorem csInf_sub {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddBelow s) (ht₀ : Set.Nonempty t) (ht₁ : BddAbove t) :
sInf (s - t) = sInf s - sSup t
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theorem csInf_div {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] {s : Set α} {t : Set α} (hs₀ : Set.Nonempty s) (hs₁ : BddBelow s) (ht₀ : Set.Nonempty t) (ht₁ : BddAbove t) :
sInf (s / t) = sInf s / sSup t
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theorem LinearOrderedField.smul_Ioo {K : Type u_2} [LinearOrderedField K] {a : K} {b : K} {r : K} (hr : 0 < r) :
r • Set.Ioo a b = Set.Ioo (r • a) (r • b)
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theorem LinearOrderedField.smul_Icc {K : Type u_2} [LinearOrderedField K] {a : K} {b : K} {r : K} (hr : 0 < r) :
r • Set.Icc a b = Set.Icc (r • a) (r • b)
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theorem LinearOrderedField.smul_Ico {K : Type u_2} [LinearOrderedField K] {a : K} {b : K} {r : K} (hr : 0 < r) :
r • Set.Ico a b = Set.Ico (r • a) (r • b)
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theorem LinearOrderedField.smul_Ioc {K : Type u_2} [LinearOrderedField K] {a : K} {b : K} {r : K} (hr : 0 < r) :
r • Set.Ioc a b = Set.Ioc (r • a) (r • b)
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theorem LinearOrderedField.smul_Ioi {K : Type u_2} [LinearOrderedField K] {a : K} {r : K} (hr : 0 < r) :
r • Set.Ioi a = Set.Ioi (r • a)
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theorem LinearOrderedField.smul_Iio {K : Type u_2} [LinearOrderedField K] {a : K} {r : K} (hr : 0 < r) :
r • Set.Iio a = Set.Iio (r • a)
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theorem LinearOrderedField.smul_Ici {K : Type u_2} [LinearOrderedField K] {a : K} {r : K} (hr : 0 < r) :
r • Set.Ici a = Set.Ici (r • a)
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theorem LinearOrderedField.smul_Iic {K : Type u_2} [LinearOrderedField K] {a : K} {r : K} (hr : 0 < r) :
r • Set.Iic a = Set.Iic (r • a)