Loogle!

Result

Found 449 declarations mentioning Pairwise. Of these, only the first 200 are shown.

• Pairwise 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} (r : α → α → Prop) : Prop
• Subsingleton.pairwise 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {r : α → α → Prop} [Subsingleton α] : Pairwise r
• Pairwise.set_pairwise 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {r : α → α → Prop} (h : Pairwise r) (s : Set α) : s.Pairwise r
• Pairwise.eq 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {r : α → α → Prop} {a b : α} (h : Pairwise r) : ¬r a b → a = b
• Pairwise.mono 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {r p : α → α → Prop} (hr : Pairwise r) (h : ∀ ⦃i j : α⦄, r i j → p i j) : Pairwise p
• Function.Injective.pairwise_ne 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {ι : Type u_3} {f : ι → α} : Function.Injective f → Pairwise (Function.onFun (fun x1 x2 => x1 ≠ x2) f)
• Function.injective_iff_pairwise_ne 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {ι : Type u_3} {f : ι → α} : Function.Injective f ↔ Pairwise (Function.onFun (fun x1 x2 => x1 ≠ x2) f)
• Pairwise.comp_of_injective 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} (hr : Pairwise r) {f : β → α} (hf : Function.Injective f) : Pairwise (Function.onFun r f)
• Pairwise.of_comp_of_surjective 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : β → α} (hr : Pairwise (Function.onFun r f)) (hf : Function.Surjective f) : Pairwise r
• Function.Bijective.pairwise_comp_iff 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : β → α} (hf : Function.Bijective f) : Pairwise (Function.onFun r f) ↔ Pairwise r
• Set.Pairwise.on_injective 📋 Mathlib.Logic.Pairwise
  {α : Type u_1} {ι : Type u_3} {r : α → α → Prop} {f : ι → α} {s : Set α} (hs : s.Pairwise r) (hf : Function.Injective f) (hfs : ∀ (x : ι), f x ∈ s) : Pairwise (Function.onFun r f)
• pairwise_fin_succ_iff_of_isSymm 📋 Mathlib.Logic.Pairwise
  {n : ℕ} {R : Fin n.succ → Fin n.succ → Prop} [Std.Symm R] : Pairwise R ↔ (∀ (j : Fin n), R 0 j.succ) ∧ Pairwise fun i j => R i.succ j.succ
• pairwise_fin_succ_iff 📋 Mathlib.Logic.Pairwise
  {n : ℕ} {R : Fin n.succ → Fin n.succ → Prop} : Pairwise R ↔ (∀ (i : Fin n), R i.succ 0) ∧ (∀ (j : Fin n), R 0 j.succ) ∧ Pairwise fun i j => R i.succ j.succ
• Pairwise.of_gt 📋 Mathlib.Order.Lattice
  {α : Type u} [LinearOrder α] {p : α → α → Prop} [Std.Symm p] : (∀ ⦃a b : α⦄, b < a → p a b) → Pairwise p
• Pairwise.of_lt 📋 Mathlib.Order.Lattice
  {α : Type u} [LinearOrder α] {p : α → α → Prop} [Std.Symm p] : (∀ ⦃a b : α⦄, a < b → p a b) → Pairwise p
• pairwise_iff_gt 📋 Mathlib.Order.Lattice
  {α : Type u} [LinearOrder α] {p : α → α → Prop} [Std.Symm p] : Pairwise p ↔ ∀ ⦃a b : α⦄, b < a → p a b
• pairwise_iff_lt 📋 Mathlib.Order.Lattice
  {α : Type u} [LinearOrder α] {p : α → α → Prop} [Std.Symm p] : Pairwise p ↔ ∀ ⦃a b : α⦄, a < b → p a b
• biSup_iInter_of_pairwise_disjoint 📋 Mathlib.Order.CompleteBooleanAlgebra
  {α : Type u} [CompletelyDistribLattice α] {ι : Type u_1} {κ : Type u_2} [hκ : Nonempty κ] {f : ι → α} (h : Pairwise (Function.onFun Disjoint f)) (s : κ → Set ι) : ⨆ i ∈ ⋂ j, s j, f i = ⨅ j, ⨆ i ∈ s j, f i
• biInf_inter_of_pairwise_codisjoint 📋 Mathlib.Order.CompleteBooleanAlgebra
  {α : Type u} [Order.Coframe α] {ι : Type u_1} {f : ι → α} (h : Pairwise (Function.onFun Codisjoint f)) (s t : Set ι) : ⨅ i ∈ s ∩ t, f i = (⨅ i ∈ s, f i) ⊔ ⨅ i ∈ t, f i
• biSup_inter_of_pairwise_disjoint 📋 Mathlib.Order.CompleteBooleanAlgebra
  {α : Type u} [Order.Frame α] {ι : Type u_1} {f : ι → α} (h : Pairwise (Function.onFun Disjoint f)) (s t : Set ι) : ⨆ i ∈ s ∩ t, f i = (⨆ i ∈ s, f i) ⊓ ⨆ i ∈ t, f i
• Set.pairwise_univ 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {r : α → α → Prop} : Set.univ.Pairwise r ↔ Pairwise r
• pairwise_ne_iff_injective 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {f : ι → α} : (Pairwise fun i j => f i ≠ f j) ↔ Function.Injective f
• pairwise_not_eq_iff_injective 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {f : ι → α} : (Pairwise fun i j => ¬f i = f j) ↔ Function.Injective f
• Pairwise.range_pairwise 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {r : α → α → Prop} {f : ι → α} (hr : Pairwise (Function.onFun r f)) : (Set.range f).Pairwise r
• pairwise_on_bool 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {r : α → α → Prop} [Std.Symm r] {a b : α} : Pairwise (Function.onFun r fun c => bif c then a else b) ↔ r a b
• Pairwise.pairwiseDisjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {f : ι → α} (h : Pairwise (Function.onFun Disjoint f)) (s : Set ι) : s.PairwiseDisjoint f
• pairwise_disjoint_on_bool 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} : Pairwise (Function.onFun Disjoint fun c => bif c then a else b) ↔ Disjoint a b
• Pairwise.set_of_subtype 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} (s : Set α) (r : α → α → Prop) : (Pairwise fun x y => r ↑x ↑y) → s.Pairwise r
• Symmetric.pairwise_on 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {r : α → α → Prop} [LinearOrder ι] [Std.Symm r] (f : ι → α) : Pairwise (Function.onFun r f) ↔ ∀ ⦃m n : ι⦄, m < n → r (f m) (f n)
• Set.Pairwise.subtype 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} (s : Set α) (r : α → α → Prop) : s.Pairwise r → Pairwise fun x y => r ↑x ↑y
• Std.Symm.pairwise_on 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {r : α → α → Prop} [LinearOrder ι] [Std.Symm r] (f : ι → α) : Pairwise (Function.onFun r f) ↔ ∀ ⦃m n : ι⦄, m < n → r (f m) (f n)
• pairwise_subtype_iff_pairwise_set 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} (s : Set α) (r : α → α → Prop) : (Pairwise fun x y => r ↑x ↑y) ↔ s.Pairwise r
• pairwise_disjoint_mono 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {f g : ι → α} [PartialOrder α] [OrderBot α] (hs : Pairwise (Function.onFun Disjoint f)) (h : g ≤ f) : Pairwise (Function.onFun Disjoint g)
• pairwise_disjoint_on 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] [LinearOrder ι] (f : ι → α) : Pairwise (Function.onFun Disjoint f) ↔ ∀ ⦃m n : ι⦄, m < n → Disjoint (f m) (f n)
• pairwise_disjoint_fiber 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} (f : ι → α) : Pairwise (Function.onFun Disjoint fun a => f ⁻¹' {a})
• subsingleton_setOfPred_mem_iff_pairwise_disjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {f : ι → Set α} : (∀ (a : α), {i | a ∈ f i}.Subsingleton) ↔ Pairwise (Function.onFun Disjoint f)
• subsingleton_setOf_mem_iff_pairwise_disjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {f : ι → Set α} : (∀ (a : α), {i | a ∈ f i}.Subsingleton) ↔ Pairwise (Function.onFun Disjoint f)
• Pairwise.disjoint_extend_bot 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PartialOrder γ] [OrderBot γ] {e : α → β} {f : α → γ} (hf : Pairwise (Function.onFun Disjoint f)) (he : Function.FactorsThrough f e) : Pairwise (Function.onFun Disjoint (Function.extend e f ⊥))
• exists_ne_mem_inter_of_not_pairwise_disjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {f : ι → Set α} (h : ¬Pairwise (Function.onFun Disjoint f)) : ∃ i j, i ≠ j ∧ ∃ x, x ∈ f i ∩ f j
• exists_lt_mem_inter_of_not_pairwise_disjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [LinearOrder ι] {f : ι → Set α} (h : ¬Pairwise (Function.onFun Disjoint f)) : ∃ i j, i < j ∧ ∃ x, x ∈ f i ∩ f j
• Finset.pairwise_subtype_iff_pairwise_finset 📋 Mathlib.Data.Finset.Defs
  {α : Type u_1} {s : Finset α} (r : α → α → Prop) : Pairwise (Function.onFun r fun x => ↑x) ↔ (↑s).Pairwise r
• Finset.pairwise_subtype_iff_pairwise_finset' 📋 Mathlib.Data.Finset.Defs
  {α : Type u_1} {β : Type u_2} {s : Finset α} (r : β → β → Prop) (f : α → β) : Pairwise (Function.onFun r fun x => f ↑x) ↔ (↑s).Pairwise (Function.onFun r f)
• Finset.pairwise_cons 📋 Mathlib.Data.Finset.Insert
  {α : Type u_1} {s : Finset α} {a : α} (ha : a ∉ s) (r : α → α → Prop) : Pairwise (Function.onFun r fun a_1 => ↑a_1) ↔ Pairwise (Function.onFun r fun a => ↑a) ∧ ∀ b ∈ s, r a b ∧ r b a
• Finset.pairwise_cons' 📋 Mathlib.Data.Finset.Insert
  {α : Type u_1} {β : Type u_2} {s : Finset α} {a : α} (ha : a ∉ s) (r : β → β → Prop) (f : α → β) : Pairwise (Function.onFun r fun a_1 => f ↑a_1) ↔ Pairwise (Function.onFun r fun a => f ↑a) ∧ ∀ b ∈ s, r (f a) (f b) ∧ r (f b) (f a)
• exists_seq_of_forall_finset_exists' 📋 Mathlib.Data.Fintype.Basic
  {α : Type u_3} (P : α → Prop) (r : α → α → Prop) [Std.Symm r] (h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) : ∃ f, (∀ (n : ℕ), P (f n)) ∧ Pairwise (Function.onFun r f)
• Set.biUnion_iInter_of_pairwise_disjoint 📋 Mathlib.Data.Set.Lattice.Image
  {α : Type u_1} {ι : Type u_9} {κ : Type u_10} [hκ : Nonempty κ] {f : ι → Set α} (h : Pairwise (Function.onFun Disjoint f)) (s : κ → Set ι) : ⋃ i ∈ ⋂ j, s j, f i = ⋂ j, ⋃ i ∈ s j, f i
• Set.biUnion_inter_of_pairwise_disjoint 📋 Mathlib.Data.Set.Lattice.Image
  {α : Type u_1} {ι : Type u_9} {f : ι → Set α} (h : Pairwise (Function.onFun Disjoint f)) (s t : Set ι) : ⋃ i ∈ s ∩ t, f i = (⋃ i ∈ s, f i) ∩ ⋃ i ∈ t, f i
• Pi.mulSingle_commute 📋 Mathlib.Algebra.Group.Pi.Lemmas
  {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → MulOneClass (f i)] : Pairwise fun i j => ∀ (x : f i) (y : f j), Commute (Pi.mulSingle i x) (Pi.mulSingle j y)
• Pi.single_addCommute 📋 Mathlib.Algebra.Group.Pi.Lemmas
  {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] : Pairwise fun i j => ∀ (x : f i) (y : f j), AddCommute (Pi.single i x) (Pi.single j y)
• Set.pairwise_disjoint_smul_iff 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s : Set β} : Pairwise (Function.onFun Disjoint fun a => a • s) ↔ ∀ (a : α), (a • s ∩ s).Nonempty → a = 1
• Set.pairwise_disjoint_vadd_iff 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s : Set β} : Pairwise (Function.onFun Disjoint fun a => a +ᵥ s) ↔ ∀ (a : α), ((a +ᵥ s) ∩ s).Nonempty → a = 0
• Function.Embedding.sigmaSet 📋 Mathlib.Logic.Embedding.Set
  {α : Type u_1} {ι : Type u_2} {s : ι → Set α} (h : Pairwise (Function.onFun Disjoint s)) : (i : ι) × ↑(s i) ↪ α
• Function.Embedding.sigmaSet_range 📋 Mathlib.Logic.Embedding.Set
  {α : Type u_1} {ι : Type u_2} {s : ι → Set α} (h : Pairwise (Function.onFun Disjoint s)) : Set.range ⇑(Function.Embedding.sigmaSet h) = ⋃ i, s i
• Function.Embedding.sigmaSet_apply 📋 Mathlib.Logic.Embedding.Set
  {α : Type u_1} {ι : Type u_2} {s : ι → Set α} (h : Pairwise (Function.onFun Disjoint s)) (x : (i : ι) × ↑(s i)) : (Function.Embedding.sigmaSet h) x = ↑x.snd
• Function.Embedding.coe_sigmaSet 📋 Mathlib.Logic.Embedding.Set
  {α : Type u_1} {ι : Type u_2} {s : ι → Set α} (h : Pairwise (Function.onFun Disjoint s)) : ⇑(Function.Embedding.sigmaSet h) = fun x => ↑x.snd
• Function.Embedding.sigmaSet_preimage 📋 Mathlib.Logic.Embedding.Set
  {α : Type u_1} {ι : Type u_2} {s : ι → Set α} (h : Pairwise (Function.onFun Disjoint s)) (i : ι) (r : Set α) : Subtype.val '' Sigma.mk i ⁻¹' ⇑(Function.Embedding.sigmaSet h) ⁻¹' r = r ∩ s i
• Multiset.pairwise_disjoint_powersetCard 📋 Mathlib.Data.Multiset.Powerset
  {α : Type u_1} (s : Multiset α) : Pairwise fun i j => Disjoint (Multiset.powersetCard i s) (Multiset.powersetCard j s)
• Set.unionEqSigmaOfDisjoint 📋 Mathlib.Data.Set.Lattice.Disjoint
  {α : Type u_1} {β : Type u_2} {t : α → Set β} (h : Pairwise (Function.onFun Disjoint t)) : ↑(⋃ i, t i) ≃ (i : α) × ↑(t i)
• Set.sigmaToiUnion_bijective 📋 Mathlib.Data.Set.Lattice.Disjoint
  {α : Type u_1} {β : Type u_2} (t : α → Set β) (h : Pairwise (Function.onFun Disjoint t)) : Function.Bijective (Set.sigmaToiUnion t)
• Set.sigmaToiUnion_injective 📋 Mathlib.Data.Set.Lattice.Disjoint
  {α : Type u_1} {β : Type u_2} (t : α → Set β) (h : Pairwise (Function.onFun Disjoint t)) : Function.Injective (Set.sigmaToiUnion t)
• Set.biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ 📋 Mathlib.Data.Set.Lattice.Disjoint
  {α : Type u_1} {ι : Type u_5} {Es : ι → Set α} (Es_union : ⋃ i, Es i = Set.univ) (Es_disj : Pairwise fun i j => Disjoint (Es i) (Es j)) (I : Set ι) : (⋃ i ∈ I, Es i)ᶜ = ⋃ i ∈ Iᶜ, Es i
• Set.coe_unionEqSigmaOfDisjoint_symm_apply 📋 Mathlib.Data.Set.Lattice.Disjoint
  {α : Type u_5} {β : Type u_6} {t : α → Set β} (h : Pairwise (Function.onFun Disjoint t)) (x : (i : α) × ↑(t i)) : ↑((Set.unionEqSigmaOfDisjoint h).symm x) = ↑x.snd
• Set.coe_snd_unionEqSigmaOfDisjoint 📋 Mathlib.Data.Set.Lattice.Disjoint
  {α : Type u_5} {β : Type u_6} {t : α → Set β} (h : Pairwise (Function.onFun Disjoint t)) (x : ↑(⋃ i, t i)) : ↑((Set.unionEqSigmaOfDisjoint h) x).snd = ↑x
• Pairwise.biUnion_injective 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {f : ι → Set α} (h₀ : Pairwise (Function.onFun Disjoint f)) (h₁ : ∀ (i : ι), (f i).Nonempty) : Function.Injective fun s => ⋃ i ∈ s, f i
• Pairwise.subset_of_biUnion_subset_biUnion 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {f : ι → Set α} {s t : Set ι} (h₀ : Pairwise (Function.onFun Disjoint f)) (h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t
• Finset.pairwise_disjoint_powersetCard 📋 Mathlib.Data.Finset.Powerset
  {α : Type u_1} (s : Finset α) : Pairwise fun i j => Disjoint (Finset.powersetCard i s) (Finset.powersetCard j s)
• Set.finite_iUnion_iff 📋 Mathlib.Data.Set.Finite.Lattice
  {α : Type u} {ι : Type u_1} {s : ι → Set α} (hs : Pairwise fun i j => Disjoint (s i) (s j)) : (⋃ i, s i).Finite ↔ (∀ (i : ι), (s i).Finite) ∧ {i | (s i).Nonempty}.Finite
• Set.countable_ofPred_nonempty_of_disjoint 📋 Mathlib.Data.Set.Countable
  {α : Type u} {β : Type v} {f : β → Set α} (hf : Pairwise (Function.onFun Disjoint f)) {s : Set α} (h'f : ∀ (t : β), f t ⊆ s) (hs : s.Countable) : {t | (f t).Nonempty}.Countable
• Set.countable_setOf_nonempty_of_disjoint 📋 Mathlib.Data.Set.Countable
  {α : Type u} {β : Type v} {f : β → Set α} (hf : Pairwise (Function.onFun Disjoint f)) {s : Set α} (h'f : ∀ (t : β), f t ⊆ s) (hs : s.Countable) : {t | (f t).Nonempty}.Countable
• Cardinal.mk_iUnion_eq_sum_mk 📋 Mathlib.SetTheory.Cardinal.Basic
  {α ι : Type u} {f : ι → Set α} (h : Pairwise (Function.onFun Disjoint f)) : Cardinal.mk ↑(⋃ i, f i) = Cardinal.sum fun i => Cardinal.mk ↑(f i)
• Cardinal.mk_iUnion_eq_sum_mk_lift 📋 Mathlib.SetTheory.Cardinal.Basic
  {α : Type u} {ι : Type v} {f : ι → Set α} (h : Pairwise (Function.onFun Disjoint f)) : Cardinal.lift.{v, u} (Cardinal.mk ↑(⋃ i, f i)) = Cardinal.sum fun i => Cardinal.mk ↑(f i)
• iSupIndep_iff_pairwiseDisjoint 📋 Mathlib.Order.SupIndep
  {α : Type u_1} {ι : Type u_3} [Order.Frame α] {f : ι → α} : iSupIndep f ↔ Pairwise (Function.onFun Disjoint f)
• iSupIndep.pairwiseDisjoint 📋 Mathlib.Order.SupIndep
  {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : iSupIndep t) : Pairwise (Function.onFun Disjoint t)
• Set.mulSupport_subset_subsingleton_of_disjoint_on_mulSupport 📋 Mathlib.Algebra.Group.Indicator
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [One β] {s : γ → Set α} (f : α → β) (hs : Pairwise (Function.onFun Disjoint fun j => s j ∩ Function.mulSupport f)) (i : α) (j : γ) (hj : i ∈ s j) : (Function.mulSupport fun d => (s d).mulIndicator f i) ⊆ {j}
• Set.support_subset_subsingleton_of_disjoint_on_support 📋 Mathlib.Algebra.Group.Indicator
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Zero β] {s : γ → Set α} (f : α → β) (hs : Pairwise (Function.onFun Disjoint fun j => s j ∩ Function.support f)) (i : α) (j : γ) (hj : i ∈ s j) : (Function.support fun d => (s d).indicator f i) ⊆ {j}
• finprod_mem_iUnion 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {ι : Type u_3} {M : Type u_5} [CommMonoid M] {f : α → M} [Finite ι] {t : ι → Set α} (h : Pairwise (Function.onFun Disjoint t)) (ht : ∀ (i : ι), (t i).Finite) : ∏ᶠ (a : α) (_ : a ∈ ⋃ i, t i), f a = ∏ᶠ (i : ι) (a : α) (_ : a ∈ t i), f a
• finsum_mem_iUnion 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {ι : Type u_3} {M : Type u_5} [AddCommMonoid M] {f : α → M} [Finite ι] {t : ι → Set α} (h : Pairwise (Function.onFun Disjoint t)) (ht : ∀ (i : ι), (t i).Finite) : ∑ᶠ (a : α) (_ : a ∈ ⋃ i, t i), f a = ∑ᶠ (i : ι) (a : α) (_ : a ∈ t i), f a
• partialSups_disjoint_of_disjoint 📋 Mathlib.Order.PartialSups
  {α : Type u_1} {ι : Type u_3} [Preorder ι] [LocallyFiniteOrderBot ι] [DistribLattice α] [OrderBot α] (f : ι → α) (h : Pairwise (Function.onFun Disjoint f)) {i j : ι} (hij : i < j) : Disjoint ((partialSups f) i) (f j)
• disjoint_disjointed 📋 Mathlib.Order.Disjointed
  {α : Type u_1} {ι : Type u_2} [GeneralizedBooleanAlgebra α] [LinearOrder ι] [LocallyFiniteOrderBot ι] (f : ι → α) : Pairwise (Function.onFun Disjoint (disjointed f))
• Fintype.exists_disjointed_le 📋 Mathlib.Order.Disjointed
  {α : Type u_1} [GeneralizedBooleanAlgebra α] {ι : Type u_3} [Fintype ι] (f : ι → α) : ∃ g ≤ f, Finset.univ.sup g = Finset.univ.sup f ∧ Pairwise (Function.onFun Disjoint g)
• disjointed_unique' 📋 Mathlib.Order.Disjointed
  {α : Type u_1} {ι : Type u_2} [GeneralizedBooleanAlgebra α] [LinearOrder ι] [LocallyFiniteOrderBot ι] {f d : ι → α} (hdisj : Pairwise (Function.onFun Disjoint d)) (hsups : partialSups d = partialSups f) : d = disjointed f
• LinearIndependent.of_pairwise_dual_eq_zero_one 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
  {ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] (v : ι → M) (f : ι → Module.Dual R M) (h1 : Pairwise fun i j => (f i) (v j) = 0) (h2 : ∀ (i : ι), (f i) (v i) = 1) : LinearIndependent R v
• Fintype.prod_dvd_of_isRelPrime 📋 Mathlib.RingTheory.Coprime.Lemmas
  {α : Type u_2} {I : Type u_1} [CommMonoid α] [DecompositionMonoid α] {z : α} {s : I → α} [Fintype I] (Hs : Pairwise (Function.onFun IsRelPrime s)) (Hs1 : ∀ (i : I), s i ∣ z) : ∏ x, s x ∣ z
• Fintype.prod_dvd_of_coprime 📋 Mathlib.RingTheory.Coprime.Lemmas
  {R : Type u} {I : Type v} [CommSemiring R] {z : R} {s : I → R} [Fintype I] (Hs : Pairwise (Function.onFun IsCoprime s)) (Hs1 : ∀ (i : I), s i ∣ z) : ∏ x, s x ∣ z
• exists_sum_eq_one_iff_pairwise_coprime' 📋 Mathlib.RingTheory.Coprime.Lemmas
  {R : Type u} {I : Type v} [CommSemiring R] {s : I → R} [Fintype I] [Nonempty I] [DecidableEq I] : (∃ μ, ∑ i, μ i * ∏ j ∈ {i}ᶜ, s j = 1) ↔ Pairwise (Function.onFun IsCoprime s)
• pairwise_coprime_iff_coprime_prod 📋 Mathlib.RingTheory.Coprime.Lemmas
  {R : Type u} {I : Type v} [CommSemiring R] {s : I → R} {t : Finset I} [DecidableEq I] : Pairwise (Function.onFun IsCoprime fun i => s ↑i) ↔ ∀ i ∈ t, IsCoprime (s i) (∏ j ∈ t \ {i}, s j)
• pairwise_isRelPrime_iff_isRelPrime_prod 📋 Mathlib.RingTheory.Coprime.Lemmas
  {α : Type u_2} {I : Type u_1} [CommMonoid α] [DecompositionMonoid α] {s : I → α} {t : Finset I} [DecidableEq I] : Pairwise (Function.onFun IsRelPrime fun i => s ↑i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j)
• exists_sum_eq_one_iff_pairwise_coprime 📋 Mathlib.RingTheory.Coprime.Lemmas
  {R : Type u} {I : Type v} [CommSemiring R] {s : I → R} {t : Finset I} [DecidableEq I] (h : t.Nonempty) : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (Function.onFun IsCoprime fun i => s ↑i)
• Ideal.iInf_span_singleton_natCast 📋 Mathlib.RingTheory.Ideal.Operations
  {R : Type u_2} [CommRing R] {ι : Type u_3} [Fintype ι] {I : ι → ℕ} (hI : Pairwise fun i j => (I i).Coprime (I j)) : ⨅ i, Ideal.span {↑(I i)} = Ideal.span {↑(∏ i, I i)}
• Matrix.mem_range_scalar_of_commute_single 📋 Mathlib.Data.Matrix.Basis
  {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} (hM : Pairwise fun i j => Commute (Matrix.single i j 1) M) : M ∈ Set.range ⇑(Matrix.scalar n)
• Ideal.exists_forall_sub_mem_ideal 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hI : Pairwise (Function.onFun IsCoprime I)) (x : ι → R) : ∃ r, ∀ (i : ι), r - x i ∈ I i
• Ideal.pi_quotient_surjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hf : Pairwise (Function.onFun IsCoprime I)) (x : (i : ι) → R ⧸ I i) : ∃ r, ∀ (i : ι), (Ideal.Quotient.mk (I i)) r = x i
• Ideal.quotientInfRingEquivPiQuotient 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] (f : ι → Ideal R) (hf : Pairwise (Function.onFun IsCoprime f)) : R ⧸ ⨅ i, f i ≃+* ((i : ι) → R ⧸ f i)
• Ideal.pi_mkQ_surjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hI : Pairwise (Function.onFun IsCoprime I)) : Function.Surjective ⇑(LinearMap.pi fun i => Submodule.mkQ (I i))
• Ideal.quotientInfToPiQuotient_surj 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hI : Pairwise (Function.onFun IsCoprime I)) : Function.Surjective ⇑(Ideal.quotientInfToPiQuotient I)
• Nat.pairwise_coprime_pow_primeFactors_factorization 📋 Mathlib.Data.Nat.Factorization.Basic
  {n : ℕ} : Pairwise (Function.onFun Nat.Coprime fun p => ↑p ^ n.factorization ↑p)
• Polynomial.pairwise_coprime_X_sub_C 📋 Mathlib.Algebra.Polynomial.RingDivision
  {K : Type u_1} [Field K] {I : Type v} {s : I → K} (H : Function.Injective s) : Pairwise (Function.onFun IsCoprime fun i => Polynomial.X - Polynomial.C (s i))
• Pairwise.exists_mem_filter_of_disjoint 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Type u_1} [Finite ι] {l : ι → Filter α} (hd : Pairwise (Function.onFun Disjoint l)) : ∃ s, (∀ (i : ι), s i ∈ l i) ∧ Pairwise (Function.onFun Disjoint s)
• Pairwise.exists_mem_filter_basis_of_disjoint 📋 Mathlib.Order.Filter.Bases.Finite
  {α : Type u_1} {I : Type u_5} [Finite I] {l : I → Filter α} {ι : I → Sort u_4} {p : (i : I) → ι i → Prop} {s : (i : I) → ι i → Set α} (hd : Pairwise (Function.onFun Disjoint l)) (h : ∀ (i : I), (l i).HasBasis (p i) (s i)) : ∃ ind, (∀ (i : I), p i (ind i)) ∧ Pairwise (Function.onFun Disjoint fun i => s i (ind i))
• Equiv.Perm.signAux3_mul_and_swap 📋 Mathlib.GroupTheory.Perm.Sign
  {α : Type u} [DecidableEq α] [Finite α] (f g : Equiv.Perm α) (s : Multiset α) (hs : ∀ (x : α), x ∈ s) : (f * g).signAux3 hs = f.signAux3 hs * g.signAux3 hs ∧ Pairwise fun x y => (Equiv.swap x y).signAux3 hs = -1
• AddSubgroup.independent_of_coprime_order 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} (hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) [Finite ι] [(i : ι) → Fintype ↥(H i)] (hcoprime : Pairwise fun i j => (Fintype.card ↥(H i)).Coprime (Fintype.card ↥(H j))) : iSupIndep H
• Subgroup.independent_of_coprime_order 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} (hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) [Finite ι] [(i : ι) → Fintype ↥(H i)] (hcoprime : Pairwise fun i j => (Fintype.card ↥(H i)).Coprime (Fintype.card ↥(H j))) : iSupIndep H
• AddSubgroup.noncommPiCoprod_range 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y} : (AddSubgroup.noncommPiCoprod hcomm).range = ⨆ i, H i
• Subgroup.noncommPiCoprod_range 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y} : (Subgroup.noncommPiCoprod hcomm).range = ⨆ i, H i
• AddMonoidHom.noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y)) : ((i : ι) → N i) →+ M
• MonoidHom.noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) (hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)) : ((i : ι) → N i) →* M
• AddSubgroup.noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} [Fintype ι] (hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) : ((i : ι) → ↥(H i)) →+ G
• Subgroup.noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} [Fintype ι] (hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) : ((i : ι) → ↥(H i)) →* G
• AddMonoidHom.noncommPiCoprodEquiv 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] [DecidableEq ι] : { ϕ // Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y) } ≃ (((i : ι) → N i) →+ M)
• MonoidHom.noncommPiCoprodEquiv 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] [DecidableEq ι] : { ϕ // Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y) } ≃ (((i : ι) → N i) →* M)
• AddMonoidHom.independent_range_of_coprime_order 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → Type u_3} [(i : ι) → AddGroup (H i)] (ϕ : (i : ι) → H i →+ G) (hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), AddCommute ((ϕ i) x) ((ϕ j) y)) [Finite ι] [(i : ι) → Fintype (H i)] (hcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))) : iSupIndep fun i => (ϕ i).range
• MonoidHom.independent_range_of_coprime_order 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Type u_3} [(i : ι) → Group (H i)] (ϕ : (i : ι) → H i →* G) (hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)) [Finite ι] [(i : ι) → Fintype (H i)] (hcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))) : iSupIndep fun i => (ϕ i).range
• AddMonoidHom.noncommPiCoprod_range 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → Type u_3} [(i : ι) → AddGroup (H i)] (ϕ : (i : ι) → H i →+ G) [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), AddCommute ((ϕ i) x) ((ϕ j) y)} : (AddMonoidHom.noncommPiCoprod ϕ hcomm).range = ⨆ i, (ϕ i).range
• MonoidHom.noncommPiCoprod_range 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Type u_3} [(i : ι) → Group (H i)] (ϕ : (i : ι) → H i →* G) [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)} : (MonoidHom.noncommPiCoprod ϕ hcomm).range = ⨆ i, (ϕ i).range
• AddMonoidHom.noncommPiCoprod_single 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y)} [DecidableEq ι] (i : ι) (y : N i) : (AddMonoidHom.noncommPiCoprod ϕ hcomm) (Pi.single i y) = (ϕ i) y
• MonoidHom.noncommPiCoprod_mulSingle 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)} [DecidableEq ι] (i : ι) (y : N i) : (MonoidHom.noncommPiCoprod ϕ hcomm) (Pi.mulSingle i y) = (ϕ i) y
• AddMonoidHom.addCommute_noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y)} {m : M} (comm : ∀ (i : ι) (x : N i), AddCommute m ((ϕ i) x)) (h : (i : ι) → N i) : AddCommute m ((AddMonoidHom.noncommPiCoprod ϕ hcomm) h)
• MonoidHom.commute_noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)} {m : M} (comm : ∀ (i : ι) (x : N i), Commute m ((ϕ i) x)) (h : (i : ι) → N i) : Commute m ((MonoidHom.noncommPiCoprod ϕ hcomm) h)
• AddMonoidHom.noncommPiCoprod_mrange 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y)} : AddMonoidHom.mrange (AddMonoidHom.noncommPiCoprod ϕ hcomm) = ⨆ i, AddMonoidHom.mrange (ϕ i)
• MonoidHom.noncommPiCoprod_mrange 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)} : MonoidHom.mrange (MonoidHom.noncommPiCoprod ϕ hcomm) = ⨆ i, MonoidHom.mrange (ϕ i)
• AddSubgroup.injective_noncommPiCoprod_of_iSupIndep 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y} (hind : iSupIndep H) : Function.Injective ⇑(AddSubgroup.noncommPiCoprod hcomm)
• Subgroup.injective_noncommPiCoprod_of_iSupIndep 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y} (hind : iSupIndep H) : Function.Injective ⇑(Subgroup.noncommPiCoprod hcomm)
• AddMonoidHom.injective_noncommPiCoprod_of_iSupIndep 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → Type u_3} [(i : ι) → AddGroup (H i)] (ϕ : (i : ι) → H i →+ G) [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), AddCommute ((ϕ i) x) ((ϕ j) y)} (hind : iSupIndep fun i => (ϕ i).range) (hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)) : Function.Injective ⇑(AddMonoidHom.noncommPiCoprod ϕ hcomm)
• MonoidHom.injective_noncommPiCoprod_of_iSupIndep 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Type u_3} [(i : ι) → Group (H i)] (ϕ : (i : ι) → H i →* G) [Fintype ι] {hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)} (hind : iSupIndep fun i => (ϕ i).range) (hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)) : Function.Injective ⇑(MonoidHom.noncommPiCoprod ϕ hcomm)
• AddMonoidHom.noncommPiCoprod_apply 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y)} (h : (i : ι) → N i) : (AddMonoidHom.noncommPiCoprod ϕ hcomm) h = Finset.univ.noncommSum (fun i => (ϕ i) (h i)) ⋯
• MonoidHom.noncommPiCoprod_apply 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)} (h : (i : ι) → N i) : (MonoidHom.noncommPiCoprod ϕ hcomm) h = Finset.univ.noncommProd (fun i => (ϕ i) (h i)) ⋯
• AddSubgroup.noncommPiCoprod_single 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} [Fintype ι] [DecidableEq ι] {hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y} (i : ι) (y : ↥(H i)) : (AddSubgroup.noncommPiCoprod hcomm) (Pi.single i y) = ↑y
• Subgroup.noncommPiCoprod_mulSingle 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} [Fintype ι] [DecidableEq ι] {hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y} (i : ι) (y : ↥(H i)) : (Subgroup.noncommPiCoprod hcomm) (Pi.mulSingle i y) = ↑y
• AddSubgroup.noncommPiCoprod_apply 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} [Fintype ι] (comm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) (u : (i : ι) → ↥(H i)) : (AddSubgroup.noncommPiCoprod comm) u = Finset.univ.noncommSum (fun i => ↑(u i)) ⋯
• Subgroup.noncommPiCoprod_apply 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} [Fintype ι] (comm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) (u : (i : ι) → ↥(H i)) : (Subgroup.noncommPiCoprod comm) u = Finset.univ.noncommProd (fun i => ↑(u i)) ⋯
• AddSubgroup.addCommute_subtype_of_addCommute 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} (hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → AddCommute x y) (i j : ι) (hne : i ≠ j) (x : ↥(H i)) (y : ↥(H j)) : AddCommute ((H i).subtype x) ((H j).subtype y)
• Subgroup.commute_subtype_of_commute 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} (hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y) (i j : ι) (hne : i ≠ j) (x : ↥(H i)) (y : ↥(H j)) : Commute ((H i).subtype x) ((H j).subtype y)
• AddMonoidHom.comp_noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [AddMonoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → AddMonoid (N i)] (ϕ : (i : ι) → N i →+ M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((ϕ i) x) ((ϕ j) y)} {P : Type u_4} [AddMonoid P] {f : M →+ P} (hcomm' : Pairwise fun i j => ∀ (x : N i) (y : N j), AddCommute ((f.comp (ϕ i)) x) ((f.comp (ϕ j)) y) := ⋯) : f.comp (AddMonoidHom.noncommPiCoprod ϕ hcomm) = AddMonoidHom.noncommPiCoprod (fun i => f.comp (ϕ i)) hcomm'
• MonoidHom.comp_noncommPiCoprod 📋 Mathlib.GroupTheory.NoncommPiCoprod
  {M : Type u_1} [Monoid M] {ι : Type u_2} [Fintype ι] {N : ι → Type u_3} [(i : ι) → Monoid (N i)] (ϕ : (i : ι) → N i →* M) {hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)} {P : Type u_4} [Monoid P] {f : M →* P} (hcomm' : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((f.comp (ϕ i)) x) ((f.comp (ϕ j)) y) := ⋯) : f.comp (MonoidHom.noncommPiCoprod ϕ hcomm) = MonoidHom.noncommPiCoprod (fun i => f.comp (ϕ i)) hcomm'
• Equiv.Perm.pairwise_disjoint_of_mem_zpowers 📋 Mathlib.GroupTheory.Perm.Cycle.Factors
  {α : Type u_1} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) : Pairwise fun i j => ∀ (x y : Equiv.Perm α), x ∈ Subgroup.zpowers ↑i → y ∈ Subgroup.zpowers ↑j → x.Disjoint y
• Equiv.Perm.pairwise_commute_of_mem_zpowers 📋 Mathlib.GroupTheory.Perm.Cycle.Factors
  {α : Type u_1} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) : Pairwise fun i j => ∀ (x y : Equiv.Perm α), x ∈ Subgroup.zpowers ↑i → y ∈ Subgroup.zpowers ↑j → Commute x y
• MvPolynomial.vars_sum_of_disjoint 📋 Mathlib.Algebra.MvPolynomial.Variables
  {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (t : Finset ι) (φ : ι → MvPolynomial σ R) [DecidableEq σ] (h : Pairwise (Function.onFun Disjoint fun i => (φ i).vars)) : (∑ i ∈ t, φ i).vars = t.biUnion fun i => (φ i).vars
• Module.DualBases.eval_of_ne 📋 Mathlib.LinearAlgebra.Dual.Basis
  {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] {e : ι → M} {ε : ι → Module.Dual R M} (self : Module.DualBases e ε) : Pairwise fun i j => (ε i) (e j) = 0
• Module.DualBases.mk 📋 Mathlib.LinearAlgebra.Dual.Basis
  {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] {e : ι → M} {ε : ι → Module.Dual R M} (eval_same : ∀ (i : ι), (ε i) (e i) = 1) (eval_of_ne : Pairwise fun i j => (ε i) (e j) = 0) (total : ∀ {m₁ m₂ : M}, (∀ (i : ι), (ε i) m₁ = (ε i) m₂) → m₁ = m₂) (finite : ∀ (m : M), {i | (ε i) m ≠ 0}.Finite := by use_finite_instance) : Module.DualBases e ε
• Set.pairwise_disjoint_Ico_intCast 📋 Mathlib.Algebra.Order.Interval.Set.Group
  (α : Type u_1) [Ring α] [PartialOrder α] [IsOrderedRing α] : Pairwise (Function.onFun Disjoint fun n => Set.Ico (↑n) (↑n + 1))
• Set.pairwise_disjoint_Ioc_intCast 📋 Mathlib.Algebra.Order.Interval.Set.Group
  (α : Type u_1) [Ring α] [PartialOrder α] [IsOrderedRing α] : Pairwise (Function.onFun Disjoint fun n => Set.Ioc (↑n) (↑n + 1))
• Set.pairwise_disjoint_Ioo_intCast 📋 Mathlib.Algebra.Order.Interval.Set.Group
  (α : Type u_1) [Ring α] [PartialOrder α] [IsOrderedRing α] : Pairwise (Function.onFun Disjoint fun n => Set.Ioo (↑n) (↑n + 1))
• Set.pairwise_disjoint_Ico_zpow 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ico (b ^ n) (b ^ (n + 1)))
• Set.pairwise_disjoint_Ico_zsmul 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ico (n • b) ((n + 1) • b))
• Set.pairwise_disjoint_Ioc_zpow 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioc (b ^ n) (b ^ (n + 1)))
• Set.pairwise_disjoint_Ioc_zsmul 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioc (n • b) ((n + 1) • b))
• Set.pairwise_disjoint_Ioo_zpow 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioo (b ^ n) (b ^ (n + 1)))
• Set.pairwise_disjoint_Ioo_zsmul 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioo (n • b) ((n + 1) • b))
• Set.pairwise_disjoint_Ico_add_intCast 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [Ring α] [PartialOrder α] [IsOrderedRing α] (a : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ico (a + ↑n) (a + ↑n + 1))
• Set.pairwise_disjoint_Ioc_add_intCast 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [Ring α] [PartialOrder α] [IsOrderedRing α] (a : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioc (a + ↑n) (a + ↑n + 1))
• Set.pairwise_disjoint_Ioo_add_intCast 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [Ring α] [PartialOrder α] [IsOrderedRing α] (a : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioo (a + ↑n) (a + ↑n + 1))
• Set.pairwise_disjoint_Ico_add_zsmul 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (a b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ico (a + n • b) (a + (n + 1) • b))
• Set.pairwise_disjoint_Ico_mul_zpow 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (a b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ico (a * b ^ n) (a * b ^ (n + 1)))
• Set.pairwise_disjoint_Ioc_add_zsmul 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (a b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioc (a + n • b) (a + (n + 1) • b))
• Set.pairwise_disjoint_Ioc_mul_zpow 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (a b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioc (a * b ^ n) (a * b ^ (n + 1)))
• Set.pairwise_disjoint_Ioo_add_zsmul 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (a b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioo (a + n • b) (a + (n + 1) • b))
• Set.pairwise_disjoint_Ioo_mul_zpow 📋 Mathlib.Algebra.Order.Interval.Set.Group
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (a b : α) : Pairwise (Function.onFun Disjoint fun n => Set.Ioo (a * b ^ n) (a * b ^ (n + 1)))
• Pairwise.countable_of_isOpen_disjoint 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} (hd : Pairwise (Function.onFun Disjoint s)) (ho : ∀ (i : ι), IsOpen (s i)) (hne : ∀ (i : ι), (s i).Nonempty) : Countable ι
• Set.disjoint_accumulate 📋 Mathlib.Order.SetAccumulate
  {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] (hs : Pairwise (Function.onFun Disjoint s)) {i j : α} (hij : i < j) : Disjoint (Set.accumulate s i) (s j)
• t0Space_iff_not_inseparable 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ¬Inseparable x y
• t0Space_iff_exists_isOpen_xor'_mem 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)
• t0Space_iff_exists_isOpen_xor_mem 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)
• t1Space_iff_exists_open 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} [TopologicalSpace X] : T1Space X ↔ Pairwise fun x y => ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
• t0Space_iff_or_notMem_closure 📋 Mathlib.Topology.Separation.Basic
  (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun a b => a ∉ closure {b} ∨ b ∉ closure {a}
• pairwise_disjoint_nhds 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] [T2Space X] : Pairwise (Function.onFun Disjoint nhds)
• t2Space_iff_disjoint_nhds 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] : T2Space X ↔ Pairwise fun x y => Disjoint (nhds x) (nhds y)
• t2Space_iff_nhds 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] : T2Space X ↔ Pairwise fun x y => ∃ U ∈ nhds x, ∃ V ∈ nhds y, Disjoint U V
• T2Space.mk 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u} [TopologicalSpace X] (t2 : Pairwise fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) : T2Space X
• T2Space.t2 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u} {inst✝ : TopologicalSpace X} [self : T2Space X] : Pairwise fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
• t2Space_iff 📋 Mathlib.Topology.Separation.Hausdorff
  (X : Type u) [TopologicalSpace X] : T2Space X ↔ Pairwise fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
• subsingleton_of_disjoint_isClopen 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} {ι : Type u_1} [TopologicalSpace α] [PreconnectedSpace α] {s : ι → Set α} (h_nonempty : ∀ (i : ι), (s i).Nonempty) (h_disj : Pairwise (Function.onFun Disjoint s)) (h_clopen : ∀ (i : ι), IsClopen (s i)) : Subsingleton ι
• subsingleton_of_disjoint_isOpen_iUnion_eq_univ 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} {ι : Type u_1} [TopologicalSpace α] [PreconnectedSpace α] {s : ι → Set α} (h_nonempty : ∀ (i : ι), (s i).Nonempty) (h_disj : Pairwise (Function.onFun Disjoint s)) (h_open : ∀ (i : ι), IsOpen (s i)) (h_Union : ⋃ i, s i = Set.univ) : Subsingleton ι
• ConnectedComponents.equivOfIsClopenOfIsConnected 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} [TopologicalSpace α] {ι : Type u_3} {U : ι → Set α} (hclopen : ∀ (i : ι), IsClopen (U i)) (hdisj : Pairwise (Function.onFun Disjoint U)) (hunion : ⋃ i, U i = Set.univ) (hconn : ∀ (i : ι), IsConnected (U i)) : ConnectedComponents α ≃ ι
• subsingleton_of_disjoint_isClosed_iUnion_eq_univ 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} {ι : Type u_1} [TopologicalSpace α] [PreconnectedSpace α] {s : ι → Set α} (h_nonempty : ∀ (i : ι), (s i).Nonempty) (h_disj : Pairwise (Function.onFun Disjoint s)) [Finite ι] (h_closed : ∀ (i : ι), IsClosed (s i)) (h_Union : ⋃ i, s i = Set.univ) : Subsingleton ι
• ConnectedComponents.equivOfIsClopen 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} [TopologicalSpace α] {ι : Type u_3} {U : ι → Set α} (hclopen : ∀ (i : ι), IsClopen (U i)) (hdisj : Pairwise (Function.onFun Disjoint U)) (hunion : ⋃ i, U i = Set.univ) : ConnectedComponents α ≃ (i : ι) × ConnectedComponents ↑(U i)
• ConnectedComponents.exists_fun_isClopen_of_infinite 📋 Mathlib.Topology.Connected.Clopen
  (α : Type u) [TopologicalSpace α] [Infinite (ConnectedComponents α)] (n : ℕ) (hn : 0 < n) : ∃ U, (∀ (i : Fin n), IsClopen (U i)) ∧ (∀ (i : Fin n), (U i).Nonempty) ∧ Pairwise (Function.onFun Disjoint U) ∧ ⋃ i, U i = Set.univ
• ConnectedComponents.equivOfIsClopenOfIsConnected_mk 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} [TopologicalSpace α] {ι : Type u_3} {U : ι → Set α} (hclopen : ∀ (i : ι), IsClopen (U i)) (hdisj : Pairwise (Function.onFun Disjoint U)) (hunion : ⋃ i, U i = Set.univ) (hconn : ∀ (i : ι), IsConnected (U i)) {i : ι} (x : α) (hx : x ∈ U i) : (ConnectedComponents.equivOfIsClopenOfIsConnected hclopen hdisj hunion hconn) (ConnectedComponents.mk x) = i
• ConnectedComponents.equivOfIsClopen_mk 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} [TopologicalSpace α] {ι : Type u_3} {U : ι → Set α} (hclopen : ∀ (i : ι), IsClopen (U i)) (hdisj : Pairwise (Function.onFun Disjoint U)) (hunion : ⋃ i, U i = Set.univ) {i : ι} (x : α) (hx : x ∈ U i) : (ConnectedComponents.equivOfIsClopen hclopen hdisj hunion) (ConnectedComponents.mk x) = ⟨i, ConnectedComponents.mk ⟨x, hx⟩⟩
• ConnectedComponents.equivOfIsClopen_symm_mk 📋 Mathlib.Topology.Connected.Clopen
  {α : Type u} [TopologicalSpace α] {ι : Type u_3} {U : ι → Set α} (hclopen : ∀ (i : ι), IsClopen (U i)) (hdisj : Pairwise (Function.onFun Disjoint U)) (hunion : ⋃ i, U i = Set.univ) {i : ι} (x : ↑(U i)) : (ConnectedComponents.equivOfIsClopen hclopen hdisj hunion).symm ⟨i, ConnectedComponents.mk x⟩ = ConnectedComponents.mk ↑x
• exists_isClopen_of_totally_separated 📋 Mathlib.Topology.Connected.TotallyDisconnected
  {α : Type u_3} [TopologicalSpace α] [TotallySeparatedSpace α] : Pairwise fun x1 x2 => ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ
• totallySeparatedSpace_iff_exists_isClopen 📋 Mathlib.Topology.Connected.TotallyDisconnected
  {α : Type u_3} [TopologicalSpace α] : TotallySeparatedSpace α ↔ Pairwise fun x1 x2 => ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ
• Filter.HasBasis.uniformSpace_eq_bot 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {ι : Sort u_2} {p : ι → Prop} {s : ι → SetRel α α} {u : UniformSpace α} (h : (uniformity α).HasBasis p s) : u = ⊥ ↔ ∃ i, p i ∧ Pairwise fun x y => (x, y) ∉ s i
• t0Space_iff_uniformity' 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] : T0Space α ↔ Pairwise fun x y => ∃ r ∈ uniformity α, (x, y) ∉ r
• isClosed_range_of_spaced_out 📋 Mathlib.Topology.UniformSpace.Separation
  {α : Type u} [UniformSpace α] {ι : Type u_1} [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ uniformity α) {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (Set.range f)
• isUniformEmbedding_of_spaced_out 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {β : Type v} [UniformSpace β] {α : Type u_1} {f : α → β} {s : Set (β × β)} (hs : s ∈ uniformity β) (hf : Pairwise fun x y => (f x, f y) ∉ s) : IsUniformEmbedding f
• isClosedEmbedding_of_spaced_out 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {β : Type v} [UniformSpace β] {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α] [T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ uniformity β) (hf : Pairwise fun x y => (f x, f y) ∉ s) : Topology.IsClosedEmbedding f
• comap_uniformity_of_spaced_out 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {β : Type v} [UniformSpace β] {α : Type u_1} {f : α → β} {s : Set (β × β)} (hs : s ∈ uniformity β) (hf : Pairwise fun x y => (f x, f y) ∉ s) : Filter.comap (Prod.map f f) (uniformity β) = Filter.principal SetRel.id
• IsPreirreducible.of_subset_iUnion 📋 Mathlib.Topology.Sets.OpenCover
  {X : Type u_1} {ι : Type u_2} [TopologicalSpace X] {U : ι → TopologicalSpace.Opens X} (hn : Pairwise (Function.onFun (fun x1 x2 => ¬Disjoint x1 x2) U)) (h : ∀ (i : ι), IsPreirreducible ↑(U i)) {s : Set X} (hs : IsOpen s) (hsU : s ⊆ ⋃ i, ↑(U i)) : IsPreirreducible s
• PreirreducibleSpace.of_isOpenCover 📋 Mathlib.Topology.Sets.OpenCover
  {X : Type u_1} {ι : Type u_2} [TopologicalSpace X] {U : ι → TopologicalSpace.Opens X} (hn : Pairwise (Function.onFun (fun x1 x2 => ¬Disjoint x1 x2) U)) (hU : TopologicalSpace.IsOpenCover U) (h : ∀ (i : ι), PreirreducibleSpace ↥(U i)) : PreirreducibleSpace X
• Matrix.det_ne_zero_of_sum_col_pos 📋 Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
  {n : Type u_1} [Fintype n] [DecidableEq n] {S : Type u_2} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {A : Matrix n n S} (h1 : Pairwise fun i j => A i j < 0) (h2 : ∀ (j : n), 0 < ∑ i, A i j) : A.det ≠ 0
• Matrix.det_ne_zero_of_sum_row_pos 📋 Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
  {n : Type u_1} [Fintype n] [DecidableEq n] {S : Type u_2} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {A : Matrix n n S} (h1 : Pairwise fun i j => A i j < 0) (h2 : ∀ (i : n), 0 < ∑ j, A i j) : A.det ≠ 0
• OrthogonalIdempotents.ortho 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} [Semiring R] {I : Type u_3} {e : I → R} (self : OrthogonalIdempotents e) : Pairwise fun x1 x2 => e x1 * e x2 = 0
• OrthogonalIdempotents.mk 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} [Semiring R] {I : Type u_3} {e : I → R} (idem : ∀ (i : I), IsIdempotentElem (e i)) (ortho : Pairwise fun x1 x2 => e x1 * e x2 = 0) : OrthogonalIdempotents e
• orthogonalIdempotents_iff 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} [Semiring R] {I : Type u_3} (e : I → R) : OrthogonalIdempotents e ↔ (∀ (i : I), IsIdempotentElem (e i)) ∧ Pairwise fun x1 x2 => e x1 * e x2 = 0
• CompleteOrthogonalIdempotents.iff_ortho_complete 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} [Semiring R] {I : Type u_3} {e : I → R} [Fintype I] : CompleteOrthogonalIdempotents e ↔ (Pairwise fun x1 x2 => e x1 * e x2 = 0) ∧ ∑ i, e i = 1
• Polynomial.separable_prod 📋 Mathlib.FieldTheory.Separable
  {R : Type u} [CommRing R] {ι : Type u_1} [Fintype ι] {f : ι → Polynomial R} (h1 : Pairwise (Function.onFun IsCoprime f)) (h2 : ∀ (x : ι), (f x).Separable) : (∏ x, f x).Separable
• Module.End.iSup_iInf_maxGenEigenspace_eq_top_of_iSup_maxGenEigenspace_eq_top_of_commute 📋 Mathlib.LinearAlgebra.Eigenspace.Pi
  {ι : Type u_1} {K : Type u_3} {M : Type u_4} [Field K] [AddCommGroup M] [Module K M] [FiniteDimensional K M] (f : ι → Module.End K M) (h : Pairwise fun i j => Commute (f i) (f j)) (h' : ∀ (i : ι), ⨆ μ, (f i).maxGenEigenspace μ = ⊤) : ⨆ χ, ⨅ i, (f i).maxGenEigenspace (χ i) = ⊤
• LinearMap.BilinForm.linearIndependent_of_pairwise_le_zero 📋 Mathlib.LinearAlgebra.QuadraticForm.Dual
  {ι : Type u_4} {R : Type u_5} {M : Type u_6} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup M] [Module R M] (B : LinearMap.BilinForm R M) (hB : (LinearMap.BilinMap.toQuadraticMap B).PosDef) (f : Module.Dual R M) (v : ι → M) (hp : ∀ (i : ι), 0 < f (v i)) (hn : Pairwise fun i j => (B (v i)) (v j) ≤ 0) : LinearIndependent R v
• RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_ne 📋 Mathlib.LinearAlgebra.RootSystem.CartanMatrix
  {ι : Type u_1} {M : Type u_3} {N : Type u_4} [AddCommGroup M] [AddCommGroup N] [Finite ι] {K : Type u_6} [Field K] [CharZero K] [Module K M] [Module K N] {P : RootPairing ι K M N} [P.IsRootSystem] [P.IsCrystallographic] (b : P.Base) : ∃ d ∈ Submodule.span K (Set.range fun i j => ↑(P.pairingIn ℤ j ↑i)), (∀ (i : ι), d i ≠ 0) ∧ Pairwise (Function.onFun (fun x1 x2 => x1 ≠ x2) d)
• DirectSum.toLieAlgebra 📋 Mathlib.Algebra.Lie.DirectSum
  {R : Type u} {ι : Type v} [CommRing R] {L : ι → Type w} [(i : ι) → LieRing (L i)] [(i : ι) → LieAlgebra R (L i)] [DecidableEq ι] (L' : Type w₁) [LieRing L'] [LieAlgebra R L'] (f : (i : ι) → L i →ₗ⁅R⁆ L') (hf : Pairwise fun i j => ∀ (x : L i) (y : L j), ⁅(f i) x, (f j) y⁆ = 0) : (DirectSum ι fun i => L i) →ₗ⁅R⁆ L'

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c