Documentation

Mathlib.Algebra.Bounds

Upper/lower bounds in ordered monoids and groups #

In this file we prove a few facts like “-s is bounded above iff s is bounded below” (bddAbove_neg).

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@[simp]
theorem bddAbove_neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} :
BddAbove (-s) ↔ BddBelow s
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@[simp]
theorem bddAbove_inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} :
BddAbove s⁻¹ ↔ BddBelow s
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@[simp]
theorem bddBelow_neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} :
BddBelow (-s) ↔ BddAbove s
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@[simp]
theorem bddBelow_inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} :
BddBelow s⁻¹ ↔ BddAbove s
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theorem BddAbove.neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} (h : BddAbove s) :
BddBelow (-s)
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theorem BddAbove.inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} (h : BddAbove s) :
BddBelow s⁻¹
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theorem BddBelow.neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} (h : BddBelow s) :
BddAbove (-s)
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theorem BddBelow.inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} (h : BddBelow s) :
BddAbove s⁻¹
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@[simp]
theorem isLUB_neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsLUB (-s) a ↔ IsGLB s (-a)
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@[simp]
theorem isLUB_inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsLUB s⁻¹ a ↔ IsGLB s a⁻¹
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theorem isLUB_neg' {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsLUB (-s) (-a) ↔ IsGLB s a
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theorem isLUB_inv' {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a
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theorem IsGLB.neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} (h : IsGLB s a) :
IsLUB (-s) (-a)
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theorem IsGLB.inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} (h : IsGLB s a) :
IsLUB s⁻¹ a⁻¹
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@[simp]
theorem isGLB_neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsGLB (-s) a ↔ IsLUB s (-a)
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@[simp]
theorem isGLB_inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsGLB s⁻¹ a ↔ IsLUB s a⁻¹
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theorem isGLB_neg' {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsGLB (-s) (-a) ↔ IsLUB s a
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theorem isGLB_inv' {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} :
IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a
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theorem IsLUB.neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} (h : IsLUB s a) :
IsGLB (-s) (-a)
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theorem IsLUB.inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {s : Set G} {a : G} (h : IsLUB s a) :
IsGLB s⁻¹ a⁻¹
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theorem BddBelow.range_neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {α : Type u_2} {f : α → G} (hf : BddBelow (Set.range f)) :
BddAbove (Set.range fun (x : α) => -f x)
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theorem BddBelow.range_inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {α : Type u_2} {f : α → G} (hf : BddBelow (Set.range f)) :
BddAbove (Set.range fun (x : α) => (f x)⁻¹)
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theorem BddAbove.range_neg {G : Type u_1} [AddGroup G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] {α : Type u_2} {f : α → G} (hf : BddAbove (Set.range f)) :
BddBelow (Set.range fun (x : α) => -f x)
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theorem BddAbove.range_inv {G : Type u_1} [Group G] [Preorder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] {α : Type u_2} {f : α → G} (hf : BddAbove (Set.range f)) :
BddBelow (Set.range fun (x : α) => (f x)⁻¹)
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theorem add_mem_upperBounds_add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} {a : M} {b : M} (ha : a ∈ upperBounds s) (hb : b ∈ upperBounds t) :
a + b ∈ upperBounds (s + t)
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theorem mul_mem_upperBounds_mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} {a : M} {b : M} (ha : a ∈ upperBounds s) (hb : b ∈ upperBounds t) :
a * b ∈ upperBounds (s * t)
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theorem subset_upperBounds_add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] (s : Set M) (t : Set M) :
upperBounds s + upperBounds t ⊆ upperBounds (s + t)
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theorem subset_upperBounds_mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] (s : Set M) (t : Set M) :
upperBounds s * upperBounds t ⊆ upperBounds (s * t)
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theorem add_mem_lowerBounds_add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} {a : M} {b : M} (ha : a ∈ lowerBounds s) (hb : b ∈ lowerBounds t) :
a + b ∈ lowerBounds (s + t)
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theorem mul_mem_lowerBounds_mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} {a : M} {b : M} (ha : a ∈ lowerBounds s) (hb : b ∈ lowerBounds t) :
a * b ∈ lowerBounds (s * t)
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theorem subset_lowerBounds_add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] (s : Set M) (t : Set M) :
lowerBounds s + lowerBounds t ⊆ lowerBounds (s + t)
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theorem subset_lowerBounds_mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] (s : Set M) (t : Set M) :
lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t)
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theorem BddAbove.add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} (hs : BddAbove s) (ht : BddAbove t) :
BddAbove (s + t)
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theorem BddAbove.mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} (hs : BddAbove s) (ht : BddAbove t) :
BddAbove (s * t)
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theorem BddBelow.add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} (hs : BddBelow s) (ht : BddBelow t) :
BddBelow (s + t)
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theorem BddBelow.mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] {s : Set M} {t : Set M} (hs : BddBelow s) (ht : BddBelow t) :
BddBelow (s * t)
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theorem BddAbove.range_add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] {α : Type u_2} {f : α → M} {g : α → M} (hf : BddAbove (Set.range f)) (hg : BddAbove (Set.range g)) :
BddAbove (Set.range fun (x : α) => f x + g x)
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theorem BddAbove.range_mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] {α : Type u_2} {f : α → M} {g : α → M} (hf : BddAbove (Set.range f)) (hg : BddAbove (Set.range g)) :
BddAbove (Set.range fun (x : α) => f x * g x)
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theorem BddBelow.range_add {M : Type u_1} [Add M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1] {α : Type u_2} {f : α → M} {g : α → M} (hf : BddBelow (Set.range f)) (hg : BddBelow (Set.range g)) :
BddBelow (Set.range fun (x : α) => f x + g x)
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theorem BddBelow.range_mul {M : Type u_1} [Mul M] [Preorder M] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] [CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1] {α : Type u_2} {f : α → M} {g : α → M} (hf : BddBelow (Set.range f)) (hg : BddBelow (Set.range g)) :
BddBelow (Set.range fun (x : α) => f x * g x)
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theorem ciSup_add {ι : Type u_1} {G : Type u_2} [AddGroup G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddAbove (Set.range f)) (a : G) :
(⨆ (i : ι), f i) + a = ⨆ (i : ι), f i + a
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theorem ciSup_mul {ι : Type u_1} {G : Type u_2} [Group G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddAbove (Set.range f)) (a : G) :
(⨆ (i : ι), f i) * a = ⨆ (i : ι), f i * a
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theorem ciSup_sub {ι : Type u_1} {G : Type u_2} [AddGroup G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddAbove (Set.range f)) (a : G) :
(⨆ (i : ι), f i) - a = ⨆ (i : ι), f i - a
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theorem ciSup_div {ι : Type u_1} {G : Type u_2} [Group G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddAbove (Set.range f)) (a : G) :
(⨆ (i : ι), f i) / a = ⨆ (i : ι), f i / a
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theorem ciInf_add {ι : Type u_1} {G : Type u_2} [AddGroup G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddBelow (Set.range f)) (a : G) :
(⨅ (i : ι), f i) + a = ⨅ (i : ι), f i + a
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theorem ciInf_mul {ι : Type u_1} {G : Type u_2} [Group G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddBelow (Set.range f)) (a : G) :
(⨅ (i : ι), f i) * a = ⨅ (i : ι), f i * a
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theorem ciInf_sub {ι : Type u_1} {G : Type u_2} [AddGroup G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddBelow (Set.range f)) (a : G) :
(⨅ (i : ι), f i) - a = ⨅ (i : ι), f i - a
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theorem ciInf_div {ι : Type u_1} {G : Type u_2} [Group G] [ConditionallyCompleteLattice G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddBelow (Set.range f)) (a : G) :
(⨅ (i : ι), f i) / a = ⨅ (i : ι), f i / a
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theorem add_ciSup {ι : Sort u_1} {G : Type u_2} [AddGroup G] [ConditionallyCompleteLattice G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddAbove (Set.range f)) (a : G) :
a + ⨆ (i : ι), f i = ⨆ (i : ι), a + f i
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theorem mul_ciSup {ι : Sort u_1} {G : Type u_2} [Group G] [ConditionallyCompleteLattice G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddAbove (Set.range f)) (a : G) :
a * ⨆ (i : ι), f i = ⨆ (i : ι), a * f i
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theorem add_ciInf {ι : Sort u_1} {G : Type u_2} [AddGroup G] [ConditionallyCompleteLattice G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddBelow (Set.range f)) (a : G) :
a + ⨅ (i : ι), f i = ⨅ (i : ι), a + f i
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theorem mul_ciInf {ι : Sort u_1} {G : Type u_2} [Group G] [ConditionallyCompleteLattice G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x ≤ x_1] [Nonempty ι] {f : ι → G} (hf : BddBelow (Set.range f)) (a : G) :
a * ⨅ (i : ι), f i = ⨅ (i : ι), a * f i