Loogle!

Result

Found 187 declarations mentioning RelIso.symm.

• RelIso.symm 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : s ≃r r
• RelIso.refl_symm 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {r : α → α → Prop} : (RelIso.refl r).symm = RelIso.refl r
• RelIso.symm_bijective 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : Function.Bijective RelIso.symm
• RelIso.symm_symm 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : e.symm.symm = e
• RelIso.self_trans_symm 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : e.trans e.symm = RelIso.refl r
• RelIso.symm_trans_self 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : e.symm.trans e = RelIso.refl s
• RelIso.apply_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (x : β) : e (e.symm x) = x
• RelIso.symm_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (x : α) : e.symm (e x) = x
• RelIso.rel_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) {x : α} {y : β} : r x (e.symm y) ↔ s (e x) y
• RelIso.self_comp_symm 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : ⇑e ∘ ⇑e.symm = id
• RelIso.symm_apply_rel 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) {x : β} {y : α} : r (e.symm x) y ↔ s x (e y)
• RelIso.symm_comp_self 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : ⇑e.symm ∘ ⇑e = id
• RelIso.symm_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) (b : α) : f.symm.symm b = f b
• RelIso.apply_eq_iff_eq_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {x : α} {y : β} (f : r ≃r s) : f x = y ↔ x = f.symm y
• RelIso.cast_symm 📋 Mathlib.Order.RelIso.Basic
  {α β : Type u} {r : α → α → Prop} {s : β → β → Prop} (h₁ : α = β) (h₂ : r ≍ s) : (RelIso.cast h₁ h₂).symm = RelIso.cast ⋯ ⋯
• RelIso.eq_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) {x : β} {y : α} : y = e.symm x ↔ e y = x
• RelIso.symm_apply_eq 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) {x : β} {y : α} : e.symm x = y ↔ x = e y
• RelIso.copy 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (f : α → β) (g : β → α) (hf : f = ⇑e) (hg : g = ⇑e.symm) : r ≃r s
• RelIso.emptySumLex_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsEmpty α] (a✝ : β) : (RelIso.emptySumLex r s).symm a✝ = Sum.inr a✝
• RelIso.sumLexEmpty_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsEmpty β] (val : α) : (RelIso.sumLexEmpty r s).symm val = Sum.inl val
• RelIso.copy_eq 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (f : α → β) (g : β → α) (hf : f = ⇑e) (hg : g = ⇑e.symm) : e.copy f g hf hg = e
• RelIso.coe_copy 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (f : α → β) (g : β → α) (hf : f = ⇑e) (hg : g = ⇑e.symm) : ⇑(e.copy f g hf hg) = f
• RelIso.swap_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) (a✝ : β) : f.swap.symm a✝ = EquivLike.inv f a✝
• RelIso.symm_trans_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ≃r s) (g : s ≃r t) (a : γ) : (f.trans g).symm a = f.symm (g.symm a)
• RelIso.toEquiv_symm_ofOnFun 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (r : β → β → Prop) (f : α ≃ β) : ↑(RelIso.ofOnFun r f).symm = f.symm
• RelIso.coe_fn_symm_mk 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α ≃ β) (o : ∀ {a b : α}, s (f a) (f b) ↔ r a b) : ⇑{ toEquiv := f, map_rel_iff' := o }.symm = ⇑f.symm
• RelIso.preimage_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (f : α ≃ β) (s : β → β → Prop) (a : β) : (RelIso.preimage f s).symm a = f.symm a
• RelIso.coe_symm_ofOnFun 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (r : β → β → Prop) (f : α ≃ β) : ⇑(RelIso.ofOnFun r f).symm = ⇑f.symm
• RelIso.relIsoCongr_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₁ : r₁ ≃r s₁) : (e₁.relIsoCongr e₂) f₁ = (e₁.symm.trans f₁).trans e₂
• RelIso.relEmbeddingCongr_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₁ : r₁ ↪r s₁) : (e₁.relEmbeddingCongr e₂) f₁ = (e₁.symm.toRelEmbedding.trans f₁).trans e₂.toRelEmbedding
• RelIso.relIsoCongr_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₂ : r₂ ≃r s₂) : (e₁.relIsoCongr e₂).symm f₂ = (e₁.trans f₂).trans e₂.symm
• RelIso.relHomCongr_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₁ : r₁ →r s₁) : (e₁.relHomCongr e₂) f₁ = e₂.toRelEmbedding.toRelHom.comp (f₁.comp e₁.symm.toRelEmbedding.toRelHom)
• RelIso.relEmbeddingCongr_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₂ : r₂ ↪r s₂) : (e₁.relEmbeddingCongr e₂).symm f₂ = (e₁.toRelEmbedding.trans f₂).trans e₂.symm.toRelEmbedding
• RelIso.compl_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) (a✝ : β) : f.compl.symm a✝ = EquivLike.inv f a✝
• RelIso.toEquiv_symm_toMap 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (f : α ≃ β) : ↑(RelIso.toMap r f).symm = f.symm
• RelIso.relHomCongr_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₂ : r₂ →r s₂) : (e₁.relHomCongr e₂).symm f₂ = e₂.symm.toRelEmbedding.toRelHom.comp (f₂.comp e₁.toRelEmbedding.toRelHom)
• RelIso.coe_symm_toMap 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (f : α ≃ β) : ⇑(RelIso.toMap r f).symm = ⇑f.symm
• RelIso.relEmbeddingCongr_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₁ : r₁ ↪r s₁) (a✝ : α₂) : ((e₁.relEmbeddingCongr e₂) f₁) a✝ = e₂ (f₁ (e₁.symm a✝))
• RelIso.relHomCongr_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₁ : r₁ →r s₁) (x : α₂) : ((e₁.relHomCongr e₂) f₁) x = e₂ (f₁ (e₁.symm x))
• RelIso.relIsoCongr_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₁ : r₁ ≃r s₁) (a✝ : α₂) : ((e₁.relIsoCongr e₂) f₁) a✝ = e₂ (f₁ (e₁.symm a✝))
• RelIso.relEmbeddingCongr_symm_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₂ : r₂ ↪r s₂) (a✝ : α₁) : ((e₁.relEmbeddingCongr e₂).symm f₂) a✝ = e₂.symm (f₂ (e₁ a✝))
• RelIso.relHomCongr_symm_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₂ : r₂ →r s₂) (x : α₁) : ((e₁.relHomCongr e₂).symm f₂) x = e₂.symm (f₂ (e₁ x))
• RelIso.relIsoCongr_symm_apply_apply 📋 Mathlib.Order.RelIso.Basic
  {α₁ : Type u_5} {β₁ : Type u_6} {α₂ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {s₁ : β₁ → β₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r r₂) (e₂ : s₁ ≃r s₂) (f₂ : r₂ ≃r s₂) (a✝ : α₁) : ((e₁.relIsoCongr e₂).symm f₂) a✝ = e₂.symm (f₂ (e₁ a✝))
• RelIso.complEquiv_apply_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) (a✝ : β) : (RelIso.complEquiv f).symm a✝ = EquivLike.inv f a✝
• RelIso.complEquiv_symm_apply_symm_apply 📋 Mathlib.Order.RelIso.Basic
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : rᶜ ≃r sᶜ) (a✝ : β) : (RelIso.complEquiv.symm f).symm a✝ = EquivLike.inv f a✝
• OrderIso.toRelIsoGT_symm 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β) : e.symm.toRelIsoGT = e.toRelIsoGT.symm
• OrderIso.toRelIsoLT_symm 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β) : e.symm.toRelIsoLT = e.toRelIsoLT.symm
• OrderIso.ofUnique_symm_apply 📋 Mathlib.Order.Hom.Basic
  (α : Type u_6) (β : Type u_7) [Unique α] [Unique β] [Preorder α] [Preorder β] (a✝ : β) : (RelIso.symm (OrderIso.ofUnique α β)) a✝ = default
• OrderIso.ofRelIsoLT_symm 📋 Mathlib.Order.Hom.Basic
  {α : Type u_6} {β : Type u_7} [PartialOrder α] [PartialOrder β] (e : (fun x1 x2 => x1 < x2) ≃r fun x1 x2 => x1 < x2) : (OrderIso.ofRelIsoLT e).symm = OrderIso.ofRelIsoLT e.symm
• ULift.orderIso_symm_apply_down 📋 Mathlib.Order.Hom.Basic
  {α : Type u} [Preorder α] (down : α) : ((RelIso.symm ULift.orderIso.{v, u}) down).down = down
• OrderIso.coe_symm_toRelIsoGT 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β) : ⇑e.toRelIsoGT.symm = ⇑e.symm
• OrderIso.coe_symm_toRelIsoLT 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β) : ⇑e.toRelIsoLT.symm = ⇑e.symm
• StrictMono.orderIsoOfRightInverse_symm_apply 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [LinearOrder α] [Preorder β] (f : α → β) (h_mono : StrictMono f) (g : β → α) (hg : Function.RightInverse g f) : ⇑(RelIso.symm (StrictMono.orderIsoOfRightInverse f h_mono g hg)) = g
• OrderIso.ofSurjective_symm_apply 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [LE α] [LE β] (f : α ↪o β) (hf : Function.Surjective ⇑f) (b : β) : (RelIso.symm (OrderIso.ofSurjective f hf)) b = Function.surjInv ⋯ b
• OrderIso.prodAssoc_symm_apply 📋 Mathlib.Order.Hom.Basic
  (α : Type u_6) (β : Type u_7) (γ : Type u_8) [LE α] [LE β] [LE γ] (p : α × β × γ) : (RelIso.symm (OrderIso.prodAssoc α β γ)) p = ((p.1, p.2.1), p.2.2)
• OrderIso.arrowCongr_symm_apply 📋 Mathlib.Order.Hom.Basic
  {α : Type u_6} {β : Type u_7} {γ : Type u_8} {δ : Type u_9} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] (f : α ≃o γ) (g : β ≃o δ) (p : γ →o δ) : (RelIso.symm (f.arrowCongr g)) p = (↑g.symm).comp (p.comp ↑f)
• OrderHom.piIso_symm_apply 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} [Preorder α] {ι : Type u_6} {π : ι → Type u_7} [(i : ι) → Preorder (π i)] (f : (i : ι) → α →o π i) : (RelIso.symm OrderHom.piIso) f = OrderHom.pi f
• OrderHom.curry_symm_apply 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (f : α →o β →o γ) (x : α × β) : ((RelIso.symm OrderHom.curry) f) x = (f x.1) x.2
• OrderHom.prodIso_symm_apply 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (f : (α →o β) × (α →o γ)) : (RelIso.symm OrderHom.prodIso) f = f.1.prod f.2
• OrderIso.divRight_symm_apply 📋 Mathlib.Algebra.Order.Group.OrderIso
  {α : Type u} [Group α] [LE α] [MulRightMono α] (a b : α) : (RelIso.symm (OrderIso.divRight a)) b = b * a
• OrderIso.subRight_symm_apply 📋 Mathlib.Algebra.Order.Group.OrderIso
  {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a b : α) : (RelIso.symm (OrderIso.subRight a)) b = b + a
• OrderIso.inv_symm_apply 📋 Mathlib.Algebra.Order.Group.OrderIso
  (α : Type u) [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a✝ : αᵒᵈ) : (RelIso.symm (OrderIso.inv α)) a✝ = (OrderDual.ofDual a✝)⁻¹
• OrderIso.neg_symm_apply 📋 Mathlib.Algebra.Order.Group.OrderIso
  (α : Type u) [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a✝ : αᵒᵈ) : (RelIso.symm (OrderIso.neg α)) a✝ = -OrderDual.ofDual a✝
• OrderIso.divLeft_symm_apply 📋 Mathlib.Algebra.Order.Group.OrderIso
  {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a : α) (a✝ : αᵒᵈ) : (RelIso.symm (OrderIso.divLeft a)) a✝ = (OrderDual.ofDual a✝)⁻¹ * a
• OrderIso.subLeft_symm_apply 📋 Mathlib.Algebra.Order.Group.OrderIso
  {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a : α) (a✝ : αᵒᵈ) : (RelIso.symm (OrderIso.subLeft a)) a✝ = -OrderDual.ofDual a✝ + a
• WithZero.val_logOrderIso_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.Canonical
  {G : Type u_3} [Preorder G] [AddGroup G] (a✝ : G) : ↑((RelIso.symm WithZero.logOrderIso) a✝) = ↑(Multiplicative.ofAdd a✝)
• WithZero.expOrderIso_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.Canonical
  {G : Type u_3} [Preorder G] [AddGroup G] (a✝ : (WithZero (Multiplicative G))ˣ) : (RelIso.symm WithZero.expOrderIso) a✝ = Multiplicative.toAdd (WithZero.unzero ⋯)
• WithZero.val_inv_logOrderIso_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.Canonical
  {G : Type u_3} [Preorder G] [AddGroup G] (a✝ : G) : ↑((RelIso.symm WithZero.logOrderIso) a✝)⁻¹ = (↑(Multiplicative.ofAdd a✝))⁻¹
• OrderIso.divRight₀_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.OrderIso
  {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) (x✝ : G₀) : (RelIso.symm (OrderIso.divRight₀ a ha)) x✝ = x✝ * a
• OrderIso.mulLeft₀_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.OrderIso
  {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] (a : G₀) (ha : 0 < a) (x : G₀) : (RelIso.symm (OrderIso.mulLeft₀ a ha)) x = a⁻¹ * x
• OrderIso.mulRight₀_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.OrderIso
  {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) (x : G₀) : (RelIso.symm (OrderIso.mulRight₀ a ha)) x = x * a⁻¹
• OrderIso.compl_symm_apply 📋 Mathlib.Order.Hom.Set
  (α : Type u_1) [BooleanAlgebra α] (a✝ : αᵒᵈ) : (RelIso.symm (OrderIso.compl α)) a✝ = (OrderDual.ofDual a✝)ᶜ
• OrderIso.setCongr_symm_apply 📋 Mathlib.Order.Hom.Set
  {α : Type u_1} [Preorder α] (s t : Set α) (h : s = t) (b : { b // (fun x => x ∈ t) b }) : (RelIso.symm (s.orderIsoOfEq t h)) b = ⟨↑b, ⋯⟩
• Set.orderIsoOfEq_symm_apply 📋 Mathlib.Order.Hom.Set
  {α : Type u_1} [Preorder α] (s t : Set α) (h : s = t) (b : { b // (fun x => x ∈ t) b }) : (RelIso.symm (s.orderIsoOfEq t h)) b = ⟨↑b, ⋯⟩
• OrderIso.Set.congr_symm_apply 📋 Mathlib.Order.Hom.Set
  {α : Type u_1} [Preorder α] (s t : Set α) (h : s = t) (b : { b // (fun x => x ∈ t) b }) : (RelIso.symm (s.orderIsoOfEq t h)) b = ⟨↑b, ⋯⟩
• Fin.castOrderIso_symm_apply 📋 Mathlib.Order.Fin.Basic
  {m n : ℕ} (eq : n = m) (i : Fin m) : (RelIso.symm (Fin.castOrderIso eq)) i = Fin.cast ⋯ i
• Fin.orderIsoSubtype_symm_apply 📋 Mathlib.Order.Fin.Basic
  {n : ℕ} (a : { i // i < n }) : (RelIso.symm Fin.orderIsoSubtype) a = ⟨↑a, ⋯⟩
• OrderIso.finsetSetFinite_symm_apply 📋 Mathlib.Data.Set.Finite.Basic
  {α : Type u} (s : { s // s.Finite }) : (RelIso.symm OrderIso.finsetSetFinite) s = ⋯.toFinset
• AddSubmonoid.coe_toSubmonoid_symm_apply 📋 Mathlib.Algebra.Group.Submonoid.Operations
  {A : Type u_4} [AddZeroClass A] (S : Submonoid (Multiplicative A)) : ↑((RelIso.symm AddSubmonoid.toSubmonoid) S) = ⇑Multiplicative.ofAdd ⁻¹' ↑S
• Submonoid.coe_toAddSubmonoid_symm_apply 📋 Mathlib.Algebra.Group.Submonoid.Operations
  {M : Type u_1} [MulOneClass M] (S : AddSubmonoid (Additive M)) : ↑((RelIso.symm Submonoid.toAddSubmonoid) S) = ⇑Additive.ofMul ⁻¹' ↑S
• AddSubgroup.coe_toSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Lattice
  {A : Type u_2} [AddGroup A] (S : Subgroup (Multiplicative A)) : ↑((RelIso.symm AddSubgroup.toSubgroup) S) = ⇑Additive.toMul ⁻¹' ↑S
• Subgroup.coe_toAddSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Lattice
  {G : Type u_1} [Group G] (S : AddSubgroup (Additive G)) : ↑((RelIso.symm Subgroup.toAddSubgroup) S) = ⇑Multiplicative.toAdd ⁻¹' ↑S
• AddEquiv.comapAddSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H : Type u_4} [AddGroup H] (f : G ≃+ H) (H✝ : AddSubgroup G) : (RelIso.symm f.comapAddSubgroup) H✝ = AddSubgroup.comap (↑f.symm) H✝
• AddEquiv.mapAddSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H : Type u_5} [AddGroup H] (f : G ≃+ H) (H✝ : AddSubgroup H) : (RelIso.symm f.mapAddSubgroup) H✝ = AddSubgroup.map (↑f.symm) H✝
• MulEquiv.comapSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [Group G] {H : Type u_4} [Group H] (f : G ≃* H) (H✝ : Subgroup G) : (RelIso.symm f.comapSubgroup) H✝ = Subgroup.comap (↑f.symm) H✝
• MulEquiv.mapSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [Group G] {H : Type u_5} [Group H] (f : G ≃* H) (H✝ : Subgroup H) : (RelIso.symm f.mapSubgroup) H✝ = Subgroup.map (↑f.symm) H✝
• AddSubsemigroup.opEquiv_symm_apply 📋 Mathlib.Algebra.Group.Subsemigroup.MulOpposite
  {M : Type u_2} [Add M] (x : AddSubsemigroup Mᵃᵒᵖ) : (RelIso.symm AddSubsemigroup.opEquiv) x = x.unop
• Subsemigroup.opEquiv_symm_apply 📋 Mathlib.Algebra.Group.Subsemigroup.MulOpposite
  {M : Type u_2} [Mul M] (x : Subsemigroup Mᵐᵒᵖ) : (RelIso.symm Subsemigroup.opEquiv) x = x.unop
• AddSubmonoid.opEquiv_symm_apply 📋 Mathlib.Algebra.Group.Submonoid.MulOpposite
  {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid Mᵃᵒᵖ) : (RelIso.symm AddSubmonoid.opEquiv) x = x.unop
• Submonoid.opEquiv_symm_apply 📋 Mathlib.Algebra.Group.Submonoid.MulOpposite
  {M : Type u_2} [MulOneClass M] (x : Submonoid Mᵐᵒᵖ) : (RelIso.symm Submonoid.opEquiv) x = x.unop
• AddSubgroup.opEquiv_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.MulOpposite
  {G : Type u_1} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) : (RelIso.symm AddSubgroup.opEquiv) H = H.unop
• Subgroup.opEquiv_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.MulOpposite
  {G : Type u_1} [Group G] (H : Subgroup Gᵐᵒᵖ) : (RelIso.symm Subgroup.opEquiv) H = H.unop
• RelIso.image_eq_preimage_symm 📋 Mathlib.Order.RelIso.Set
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (t : Set α) : ⇑e '' t = ⇑e.symm ⁻¹' t
• RelIso.preimage_eq_image_symm 📋 Mathlib.Order.RelIso.Set
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (t : Set β) : ⇑e ⁻¹' t = ⇑e.symm '' t
• RelIso.subrelUnivIso_symm_apply 📋 Mathlib.Order.RelIso.Set
  {α : Type u_1} {r : α → α → Prop} {p : α → Prop} (h : ∀ (x : α), p x) (x : α) : (RelIso.subrelUnivIso h).symm x = ⟨x, ⋯⟩
• AddSubmonoid.coe_negOrderIso_symm_apply 📋 Mathlib.Algebra.Group.Submonoid.Pointwise
  {G : Type u_2} [AddGroup G] (a✝ : AddSubmonoid G) : ↑((RelIso.symm AddSubmonoid.negOrderIso) a✝) = -↑a✝
• Submonoid.coe_invOrderIso_symm_apply 📋 Mathlib.Algebra.Group.Submonoid.Pointwise
  {G : Type u_2} [Group G] (a✝ : Submonoid G) : ↑((RelIso.symm Submonoid.invOrderIso) a✝) = (↑a✝)⁻¹
• AddSubgroup.orderIsoAddCon_symm_apply_coe 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} [AddGroup G] (c : AddCon G) : ↑((RelIso.symm AddSubgroup.orderIsoAddCon) c) = c.addSubgroup
• Subgroup.orderIsoCon_symm_apply_coe 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} [Group G] (c : Con G) : ↑((RelIso.symm Subgroup.orderIsoCon) c) = c.subgroup
• Sum.Lex.toLexRelIsoLE_symm_coe 📋 Mathlib.Data.Sum.Order
  {α : Type u_1} {β : Type u_2} [LE α] [LE β] : ⇑Sum.Lex.toLexRelIsoLE.symm = ⇑ofLex
• Sum.Lex.toLexRelIsoLT_symm_coe 📋 Mathlib.Data.Sum.Order
  {α : Type u_1} {β : Type u_2} [LT α] [LT β] : ⇑Sum.Lex.toLexRelIsoLT.symm = ⇑ofLex
• Prod.Lex.prodLexAssoc_symm_apply 📋 Mathlib.Order.Hom.Lex
  (α : Type u_4) (β : Type u_5) (γ : Type u_6) [Preorder α] [Preorder β] [Preorder γ] (a✝ : Lex (α × Lex (β × γ))) : (RelIso.symm (Prod.Lex.prodLexAssoc α β γ)) a✝ = toLex (toLex ((ofLex a✝).1, (ofLex (ofLex a✝).2).1), (ofLex (ofLex a✝).2).2)
• Prod.Lex.sumLexProdLexDistrib_symm_apply 📋 Mathlib.Order.Hom.Lex
  (α : Type u_4) (β : Type u_5) (γ : Type u_6) [Preorder α] [Preorder β] [Preorder γ] (a✝ : Lex (α × γ) ⊕ₗ Lex (β × γ)) : (RelIso.symm (Prod.Lex.sumLexProdLexDistrib α β γ)) a✝ = toLex (Prod.map (⇑toLex) id ((Equiv.sumProdDistrib α β γ).symm (Sum.map (⇑ofLex) (⇑ofLex) (ofLex a✝))))
• InitialSeg.antisymm_symm 📋 Mathlib.Order.InitialSeg
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r] [IsWellOrder β s] (f : InitialSeg r s) (g : InitialSeg s r) : (f.antisymm g).symm = g.antisymm f
• PrincipalSeg.subrelIso_symm_apply 📋 Mathlib.Order.InitialSeg
  {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : PrincipalSeg r s) (a✝ : α) : f.subrelIso.symm a✝ = (Equiv.subtypeEquivProp ⋯) ⟨f.toRelEmbedding a✝, ⋯⟩
• Submodule.orderIsoMapComapOfBijective_symm_apply 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (hf : Function.Bijective ⇑f) (p : Submodule R₂ M₂) : (RelIso.symm (Submodule.orderIsoMapComapOfBijective f hf)) p = Submodule.comap f p
• Equiv.toOrderIsoSet_symm_apply 📋 Mathlib.Order.Hom.CompleteLattice
  {α : Type u_2} {β : Type u_3} (e : α ≃ β) (s : Set β) : (RelIso.symm e.toOrderIsoSet) s = ⇑e.symm '' s
• AddSubsemigroup.coe_toSubsemigroup_symm_apply 📋 Mathlib.Algebra.Group.Subsemigroup.Operations
  {A : Type u_5} [Add A] (S : Subsemigroup (Multiplicative A)) : ↑((RelIso.symm AddSubsemigroup.toSubsemigroup) S) = ⇑Multiplicative.ofAdd ⁻¹' ↑S
• Subsemigroup.coe_toAddSubsemigroup_symm_apply 📋 Mathlib.Algebra.Group.Subsemigroup.Operations
  {M : Type u_1} [Mul M] (S : AddSubsemigroup (Additive M)) : ↑((RelIso.symm Subsemigroup.toAddSubsemigroup) S) = ⇑Additive.ofMul ⁻¹' ↑S
• infIccOrderIsoIccSup_symm_apply_coe 📋 Mathlib.Order.ModularLattice
  {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (x : ↑(Set.Icc b (a ⊔ b))) : ↑((RelIso.symm (infIccOrderIsoIccSup a b)) x) = a ⊓ ↑x
• infIooOrderIsoIooSup_symm_apply_coe 📋 Mathlib.Order.ModularLattice
  {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (c : ↑(Set.Ioo b (a ⊔ b))) : ↑((RelIso.symm (infIooOrderIsoIooSup a b)) c) = a ⊓ ↑c
• infIccOrderIsoIccSup'_symm_apply_coe 📋 Mathlib.Order.ModularLattice
  {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (a✝ : ↑(Set.Icc a (a ⊔ b))) : ↑((RelIso.symm (infIccOrderIsoIccSup' a b)) a✝) = ↑((infIccOrderIsoIccSup b a).symm (((Set.Icc a (b ⊔ a)).orderIsoOfEq (Set.Icc a (a ⊔ b)) ⋯).symm a✝))
• infIooOrderIsoIooSup'_symm_apply_coe 📋 Mathlib.Order.ModularLattice
  {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (a✝ : ↑(Set.Ioo a (a ⊔ b))) : ↑((RelIso.symm (infIooOrderIsoIooSup' a b)) a✝) = ↑((infIooOrderIsoIooSup b a).symm (((Set.Ioo a (b ⊔ a)).orderIsoOfEq (Set.Ioo a (a ⊔ b)) ⋯).symm a✝))
• Submodule.orderIsoOfAlgebraMapSurjective_symm_apply 📋 Mathlib.Algebra.Algebra.Tower
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [CommRing R] [Ring S] [AddCommGroup M] [Algebra R S] [Module R M] [Module S M] [IsScalarTower R S M] (h : Function.Surjective ⇑(algebraMap R S)) (N : Submodule R M) : (RelIso.symm (Submodule.orderIsoOfAlgebraMapSurjective h)) N = { toAddSubmonoid := N.toAddSubmonoid, smul_mem' := ⋯ }
• OrderIso.smulRight_symm_apply 📋 Mathlib.Algebra.Order.Module.Defs
  {α : Type u_1} {β : Type u_2} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulMono α β] [PosSMulReflectLE α β] {a : α} (ha : 0 < a) (b : β) : (RelIso.symm (OrderIso.smulRight ha)) b = a⁻¹ • b
• Ordinal.enum_symm_apply_coe 📋 Mathlib.SetTheory.Ordinal.Basic
  {α : Type u} (r : α → α → Prop) [IsWellOrder α r] (a✝ : α) : ↑((Ordinal.enum r).symm a✝) = (Ordinal.typein r).toRelEmbedding a✝
• Ordinal.isInitialIso_symm_apply_coe 📋 Mathlib.SetTheory.Cardinal.Aleph
  (x : Cardinal.{u_1}) : ↑((RelIso.symm Ordinal.isInitialIso) x) = x.ord
• Sublattice.prodEquiv_symm_apply 📋 Mathlib.Order.Sublattice
  {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (L : Sublattice α) (M : Sublattice β) (x : { a // (fun x => x ∈ ↑L) a } × { b // (fun x => x ∈ ↑M) b }) : (RelIso.symm (L.prodEquiv M)) x = ⟨(↑x.1, ↑x.2), ⋯⟩
• Ideal.coe_piOrderIso_symm_apply 📋 Mathlib.RingTheory.Ideal.Maps
  {ι : Type u_4} {R : ι → Type u_5} [(i : ι) → Semiring (R i)] [Finite ι] (a✝ : (i : ι) → Ideal (R i)) : ↑((RelIso.symm Ideal.piOrderIso) a✝) = {r | ∀ (i : ι), r i ∈ a✝ i}
• AddCon.orderIsoOp_symm_apply 📋 Mathlib.GroupTheory.Congruence.Opposite
  {M : Type u_1} [Add M] (c : AddCon Mᵃᵒᵖ) : (RelIso.symm AddCon.orderIsoOp) c = c.unop
• Con.orderIsoOp_symm_apply 📋 Mathlib.GroupTheory.Congruence.Opposite
  {M : Type u_1} [Mul M] (c : Con Mᵐᵒᵖ) : (RelIso.symm Con.orderIsoOp) c = c.unop
• RingCon.opOrderIso_symm_apply 📋 Mathlib.RingTheory.Congruence.Opposite
  {R : Type u_1} [Add R] [Mul R] (c : RingCon Rᵐᵒᵖ) : (RelIso.symm RingCon.opOrderIso) c = c.unop
• TwoSidedIdeal.opOrderIso_symm_apply 📋 Mathlib.RingTheory.TwoSidedIdeal.Basic
  {R : Type u_1} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal Rᵐᵒᵖ) : (RelIso.symm TwoSidedIdeal.opOrderIso) I = I.unop
• TwoSidedIdeal.orderIsoRingCon_symm_apply 📋 Mathlib.RingTheory.TwoSidedIdeal.Basic
  {R : Type u_1} [NonUnitalNonAssocRing R] (ringCon : RingCon R) : (RelIso.symm TwoSidedIdeal.orderIsoRingCon) ringCon = { ringCon := ringCon }
• Subsemiring.opEquiv_symm_apply 📋 Mathlib.Algebra.Ring.Subsemiring.MulOpposite
  {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) : (RelIso.symm Subsemiring.opEquiv) S = S.unop
• Subring.opEquiv_symm_apply 📋 Mathlib.Algebra.Ring.Subring.MulOpposite
  {R : Type u_2} [NonAssocRing R] (S : Subring Rᵐᵒᵖ) : (RelIso.symm Subring.opEquiv) S = S.unop
• Subalgebra.opEquiv_symm_apply 📋 Mathlib.Algebra.Algebra.Subalgebra.MulOpposite
  {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) : (RelIso.symm Subalgebra.opEquiv) S = S.unop
• Sym2.fromRelOrderIso_symm_apply_coe 📋 Mathlib.Data.Sym.Sym2
  (α : Type u_1) (s : Set (Sym2 α)) (x y : α) : ↑((RelIso.symm (Sym2.fromRelOrderIso α)) s) x y = Sym2.ToRel s x y
• Fin.castLEOrderIso_symm_apply 📋 Mathlib.Order.Fin.Tuple
  {n m : ℕ} (h : n ≤ m) (i : { i // ↑i < n }) : (RelIso.symm (Fin.castLEOrderIso h)) i = ⟨↑↑i, ⋯⟩
• Fin.snocOrderIso_symm_apply 📋 Mathlib.Order.Fin.Tuple
  {n : ℕ} (α : Fin (n + 1) → Type u_2) [(i : Fin (n + 1)) → LE (α i)] (f : (i : Fin (n + 1)) → α i) : (RelIso.symm (Fin.snocOrderIso α)) f = (f (Fin.last n), Fin.init f)
• Fin.insertNthOrderIso_symm_apply 📋 Mathlib.Order.Fin.Tuple
  {n : ℕ} (α : Fin (n + 1) → Type u_2) [(i : Fin (n + 1)) → LE (α i)] (p : Fin (n + 1)) (f : (i : Fin (n + 1)) → α i) : (RelIso.symm (Fin.insertNthOrderIso α p)) f = (f p, p.removeNth f)
• Fin.consOrderIso_symm_apply 📋 Mathlib.Order.Fin.Tuple
  {n : ℕ} (α : Fin (n + 1) → Type u_2) [(i : Fin (n + 1)) → LE (α i)] (f : (i : Fin (n + 1)) → α i) : (RelIso.symm (Fin.consOrderIso α)) f = (f 0, Fin.tail f)
• OrderIso.asBoolAlgAsBoolRing_symm_apply 📋 Mathlib.Algebra.Ring.BooleanRing
  (α : Type u_4) [BooleanAlgebra α] (a✝ : α) : (RelIso.symm (OrderIso.asBoolAlgAsBoolRing α)) a✝ = toBoolAlg (toBoolRing a✝)
• TwoSidedIdeal.orderIsoIsTwoSided_symm_apply 📋 Mathlib.RingTheory.TwoSidedIdeal.Operations
  {R : Type u_1} [Ring R] (I : { I // I.IsTwoSided }) : (RelIso.symm TwoSidedIdeal.orderIsoIsTwoSided) I = have this := ⋯; (↑I).toTwoSided
• CategoryTheory.Subobject.mapIsoToOrderIso_symm_apply 📋 Mathlib.CategoryTheory.Subobject.Basic
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (e : X ≅ Y) (a✝ : CategoryTheory.Subobject Y) : (RelIso.symm (CategoryTheory.Subobject.mapIsoToOrderIso e)) a✝ = (CategoryTheory.Subobject.map e.inv).obj a✝
• CategoryTheory.Abelian.subobjectIsoSubobjectOp_symm_apply 📋 Mathlib.CategoryTheory.Abelian.Subobject
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] (X : C) (a : (CategoryTheory.Subobject (Opposite.op X))ᵒᵈ) : (RelIso.symm (CategoryTheory.Abelian.subobjectIsoSubobjectOp X)) a = CategoryTheory.Subobject.lift (fun x f x_1 => CategoryTheory.Subobject.mk (CategoryTheory.Limits.kernel.ι f.unop)) ⋯ a
• upperSetIsoLowerSet_symm_apply 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} [LE α] (s : LowerSet α) : (RelIso.symm upperSetIsoLowerSet) s = s.compl
• PrimeSpectrum.equivSubtype_symm_apply_asIdeal 📋 Mathlib.RingTheory.Spectrum.Prime.Defs
  (R : Type u_1) [CommSemiring R] (I : { I // I.IsPrime }) : ((RelIso.symm (PrimeSpectrum.equivSubtype R)) I).asIdeal = ↑I
• IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal 📋 Mathlib.RingTheory.Localization.Ideal
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (a✝ : { p // Disjoint ↑M ↑p.asIdeal }) : ↑((RelIso.symm (IsLocalization.primeSpectrumOrderIso M S)) a✝).asIdeal = ⋂ s, ⋂ (_ : ↑(↑a✝).asIdeal ⊆ ⇑(algebraMap R S) ⁻¹' ↑s), ↑s
• IsLocalization.orderIsoOfPrime_symm_apply_coe 📋 Mathlib.RingTheory.Localization.Ideal
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (p : { p // p.IsPrime ∧ Disjoint ↑M ↑p }) : ↑((RelIso.symm (IsLocalization.orderIsoOfPrime M S)) p) = Ideal.map (algebraMap R S) ↑p
• IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
  {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (a✝ : ↑(Set.Iic { asIdeal := I, isPrime := hI })) : ↑((RelIso.symm (IsLocalization.AtPrime.primeSpectrumOrderIso S I)) a✝).asIdeal = ⋂ s, ⋂ (_ : ↑↑((Set.orderIsoOfEq (fun p => p.IsPrime ∧ Disjoint ↑I.primeCompl ↑p) (fun p => p.IsPrime ∧ p ≤ I) ⋯).symm ⟨(↑a✝).asIdeal, ⋯⟩) ⊆ ⇑(algebraMap R S) ⁻¹' ↑s), ↑s
• IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
  {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (a✝ : { p // p.IsPrime ∧ p ≤ I }) : ↑↑((RelIso.symm (IsLocalization.AtPrime.orderIsoOfPrime S I)) a✝) = ⋂ s, ⋂ (_ : ↑↑((Set.orderIsoOfEq (fun p => p.IsPrime ∧ Disjoint ↑I.primeCompl ↑p) (fun p => p.IsPrime ∧ p ≤ I) ⋯).symm a✝) ⊆ ⇑(algebraMap R S) ⁻¹' ↑s), ↑s
• TopologicalSpace.Closeds.complOrderIso_symm_apply 📋 Mathlib.Topology.Sets.Closeds
  (α : Type u_2) [TopologicalSpace α] (a✝ : (TopologicalSpace.Opens α)ᵒᵈ) : (RelIso.symm (TopologicalSpace.Closeds.complOrderIso α)) a✝ = (TopologicalSpace.Opens.compl ∘ ⇑OrderDual.ofDual) a✝
• TopologicalSpace.Opens.complOrderIso_symm_apply 📋 Mathlib.Topology.Sets.Closeds
  (α : Type u_2) [TopologicalSpace α] (a✝ : (TopologicalSpace.Closeds α)ᵒᵈ) : (RelIso.symm (TopologicalSpace.Opens.complOrderIso α)) a✝ = (TopologicalSpace.Closeds.compl ∘ ⇑OrderDual.ofDual) a✝
• Set.powersetCard.orderIsoOfFin_symm_apply_val 📋 Mathlib.Order.Hom.PowersetCard
  {n : ℕ} {I : Type u_2} [LinearOrder I] (s : ↑(Set.powersetCard I n)) (a✝ : ↥↑s) : ↑((RelIso.symm (Set.powersetCard.orderIsoOfFin s)) a✝) = List.idxOf (↑(({x | x ∈ (↑s).sort fun x1 x2 => x1 ≤ x2}.orderIsoOfEq ↑s ⋯).symm a✝)) ((↑s).sort fun a b => a ≤ b)
• CategoryTheory.Presieve.overEquiv_symm_apply 📋 Mathlib.CategoryTheory.Sites.Over
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} (Y : CategoryTheory.Over X) (S' : CategoryTheory.Presieve Y.left) : (RelIso.symm (CategoryTheory.Presieve.overEquiv Y)) S' = CategoryTheory.Presieve.functorPullback (CategoryTheory.Over.forget X) S'
• CategoryTheory.Sieve.overEquiv_symm_apply 📋 Mathlib.CategoryTheory.Sites.Over
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} (Y : CategoryTheory.Over X) (R : CategoryTheory.Sieve ((CategoryTheory.Over.forget X).obj Y)) : (RelIso.symm (CategoryTheory.Sieve.overEquiv Y)) R = CategoryTheory.Sieve.functorPullback (CategoryTheory.Over.forget X) R
• Subfield.extendScalars.orderIso_symm_apply 📋 Mathlib.FieldTheory.IntermediateField.Basic
  {L : Type u_2} [Field L] (F : Subfield L) (E : IntermediateField (↥F) L) : (RelIso.symm (Subfield.extendScalars.orderIso F)) E = ⟨E.toSubfield, ⋯⟩
• IntermediateField.extendScalars.orderIso_symm_apply_coe 📋 Mathlib.FieldTheory.IntermediateField.Basic
  {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (F : IntermediateField K L) (E : IntermediateField (↥F) L) : ↑((RelIso.symm (IntermediateField.extendScalars.orderIso F)) E) = IntermediateField.restrictScalars K E
• TwoSidedIdeal.orderIsoMatrix_symm_apply_ringCon_r 📋 Mathlib.LinearAlgebra.Matrix.Ideal
  {R : Type u_1} {n : Type u_2} [NonAssocRing R] [Fintype n] [Nonempty n] [DecidableEq n] (J : TwoSidedIdeal (Matrix n n R)) (x y : R) : ((RelIso.symm TwoSidedIdeal.orderIsoMatrix) J).ringCon.toSetoid x y = ∀ (i j : n), J.ringCon (Matrix.single i j x) (Matrix.single i j y)
• SSet.N.orderIsoOfIso_symm_apply 📋 Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices
  {X Y : SSet} (e : X ≅ Y) (y : Y.N) : (RelIso.symm (SSet.N.orderIsoOfIso e)) y = SSet.N.mk ((CategoryTheory.ConcreteCategory.hom (e.inv.app (Opposite.op { len := y.dim }))) y.simplex) ⋯
• SSet.N.opEquiv_symm_apply 📋 Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices
  {X : SSet} (y : X.N) : (RelIso.symm SSet.N.opEquiv) y = SSet.N.mk (SSet.opObjEquiv.symm y.simplex) ⋯
• LieSubmodule.orderIsoMapComap_symm_apply 📋 Mathlib.Algebra.Lie.Submodule
  {R : Type u} {L : Type v} {M : Type w} {M' : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] [AddCommGroup M'] [Module R M'] [LieRingModule L M'] (e : M ≃ₗ⁅R,L⁆ M') (N' : LieSubmodule R L M') : (RelIso.symm (LieSubmodule.orderIsoMapComap e)) N' = LieSubmodule.comap e.toLieModuleHom N'
• OrderIso.withZeroUnits_symm_apply 📋 Mathlib.Algebra.Order.GroupWithZero.WithZero
  {α : Type u_1} [LinearOrderedCommGroupWithZero α] (a : α) : (RelIso.symm OrderIso.withZeroUnits) a = if h : a = 0 then 0 else ↑(Units.mk0 a h)
• mkFactorOrderIsoOfFactorDvdEquiv_symm_apply_coe 📋 Mathlib.RingTheory.ChainOfDivisors
  {M : Type u_1} [CommMonoidWithZero M] {N : Type u_2} [CommMonoidWithZero N] [Subsingleton Mˣ] [Subsingleton Nˣ] {m : M} {n : N} {d : { l // l ∣ m } ≃ { l // l ∣ n }} (hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l') (l : ↑(Set.Iic (Associates.mk n))) : ↑((RelIso.symm (mkFactorOrderIsoOfFactorDvdEquiv hd)) l) = Associates.mk ↑(d.symm ⟨associatesEquivOfUniqueUnits ↑l, ⋯⟩)
• PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe 📋 Mathlib.RingTheory.LocalRing.ResidueField.Fiber
  (R : Type u_3) (S : Type u_4) [CommRing R] [CommRing S] [Algebra R S] (p : Ideal R) [p.IsPrime] (a✝ : PrimeSpectrum (p.Fiber S)) : ↑↑((RelIso.symm (PrimeSpectrum.primesOverOrderIsoFiber R S p)) a✝) = ⇑Algebra.TensorProduct.includeRight ⁻¹' ↑a✝.asIdeal
• PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdeal 📋 Mathlib.RingTheory.LocalRing.ResidueField.Fiber
  (R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (p : PrimeSpectrum R) (q : PrimeSpectrum (p.asIdeal.Fiber S)) : ↑(↑((RelIso.symm (PrimeSpectrum.preimageOrderIsoFiber R S p)) q)).asIdeal = ⇑Algebra.TensorProduct.includeRight ⁻¹' ↑q.asIdeal
• OrderIso.smulRightDual_symm_apply 📋 Mathlib.Algebra.Order.Module.Field
  {𝕜 : Type u_1} (G : Type u_2) [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup G] [PartialOrder G] [IsOrderedAddMonoid G] [Module 𝕜 G] {a : 𝕜} [PosSMulMono 𝕜 G] (ha : a < 0) (a✝ : Gᵒᵈ) : (RelIso.symm (OrderIso.smulRightDual G ha)) a✝ = a⁻¹ • OrderDual.ofDual a✝
• ENNReal.mulLeftOrderIso_symm_apply 📋 Mathlib.Basic.ENNReal.Inv
  (a : ENNReal) (ha : IsUnit a) (x : ENNReal) : (RelIso.symm (a.mulLeftOrderIso ha)) x = ↑ha.unit⁻¹ * x
• ENNReal.mulRightOrderIso_symm_apply 📋 Mathlib.Basic.ENNReal.Inv
  (a : ENNReal) (ha : IsUnit a) (x : ENNReal) : (RelIso.symm (a.mulRightOrderIso ha)) x = x * ↑ha.unit⁻¹
• ValuationSubring.coe_primeSpectrumOrderEquiv_symm_apply_asIdeal 📋 Mathlib.RingTheory.Valuation.ValuationSubring
  {K : Type u} [Field K] (A : ValuationSubring K) (a✝ : { S // A ≤ S }) : ↑((RelIso.symm A.primeSpectrumOrderEquiv) a✝).asIdeal = ⇑(A.inclusion ↑a✝ ⋯) ⁻¹' ↑(IsLocalRing.maximalIdeal ↥↑a✝)
• AlgebraicGeometry.primeSpectrumOrderIsoSpec_symm_apply 📋 Mathlib.AlgebraicGeometry.Scheme
  (R : Type u) [CommRing R] (x : (↥(AlgebraicGeometry.Spec (CommRingCat.of R)))ᵒᵈ) : (RelIso.symm (AlgebraicGeometry.primeSpectrumOrderIsoSpec R)) x = OrderDual.ofDual x
• AlgebraicGeometry.specOrderIsoPrimeSpectrum_symm_apply 📋 Mathlib.AlgebraicGeometry.Scheme
  (R : CommRingCat) (x : (PrimeSpectrum ↑R)ᵒᵈ) : (RelIso.symm (AlgebraicGeometry.specOrderIsoPrimeSpectrum R)) x = OrderDual.ofDual x
• AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe 📋 Mathlib.AlgebraicGeometry.AffineScheme
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : { U // ↑U ≤ AlgebraicGeometry.Scheme.Hom.opensRange f }) : ↑((RelIso.symm (AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv f)) U) = (TopologicalSpace.Opens.map f.base).obj ↑↑U
• SSet.Subcomplex.N.orderIsoOfIso_symm_apply 📋 Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesSubcomplex
  {X : SSet} {A : X.Subcomplex} {Y : SSet} {B : Y.Subcomplex} (e : X ≅ Y) (hA : B.preimage e.hom = A) (y : B.N) : (RelIso.symm (SSet.Subcomplex.N.orderIsoOfIso e hA)) y = { toN := (SSet.N.orderIsoOfIso e).symm y.toN, notMem := ⋯ }
• SSet.Subcomplex.N.opEquiv_symm_apply 📋 Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesSubcomplex
  {X : SSet} {A : X.Subcomplex} (y : A.N) : (RelIso.symm SSet.Subcomplex.N.opEquiv) y = { toN := SSet.N.opEquiv.symm y.toN, notMem := ⋯ }
• Real.sinhOrderIso_symm_apply 📋 Mathlib.Analysis.SpecialFunctions.Arsinh
  : ⇑(RelIso.symm Real.sinhOrderIso) = Real.arsinh
• PartOrdEmb.orderIsoOfIso_symm_apply 📋 Mathlib.Order.Category.PartOrdEmb
  {α β : PartOrdEmb} (e : α ≅ β) (a : ↑β) : (RelIso.symm (PartOrdEmb.orderIsoOfIso e)) a = (CategoryTheory.ConcreteCategory.hom e.inv) a
• CategoryTheory.Subfunctor.orderIsoSubobject_symm_apply 📋 Mathlib.CategoryTheory.Subfunctor.Subobject
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor C (Type w)) (X : CategoryTheory.Subobject F) : (RelIso.symm (CategoryTheory.Subfunctor.orderIsoSubobject F)) X = CategoryTheory.Subfunctor.range X.arrow
• SimpleGraph.induceUnivIso_symm_apply_coe 📋 Mathlib.Combinatorics.SimpleGraph.Maps
  {V : Type u_1} (G : SimpleGraph V) (a : V) : ↑((RelIso.symm G.induceUnivIso) a) = a
• SimpleGraph.completeBipartiteGraphCongr_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Maps
  {V₁ : Type u_5} {V₂ : Type u_6} {W₁ : Type u_7} {W₂ : Type u_8} (hV : V₁ ≃ V₂) (hW : W₁ ≃ W₂) (a✝ : V₂ ⊕ W₂) : (RelIso.symm (SimpleGraph.completeBipartiteGraphCongr hV hW)) a✝ = Sum.map (⇑hV.symm) (⇑hW.symm) a✝
• SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Subgraph
  {V : Type u} {G : SimpleGraph V} (G' : G.Subgraph) (h : G'.IsSpanning) (v : ↑G'.verts) : (RelIso.symm (G'.spanningCoeEquivCoeOfSpanning h)) v = ↑v
• Finset.orderIsoColex_symm_apply 📋 Mathlib.Combinatorics.Colex
  (s : Colex (Finset ℕ)) : (RelIso.symm Finset.orderIsoColex) s = Finset.equivBitIndices.symm (ofColex s)
• SimpleGraph.Iso.sumComm_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Sum
  {V : Type u_3} {W : Type u_5} {G : SimpleGraph V} {H : SimpleGraph W} (a✝ : W ⊕ V) : (RelIso.symm SimpleGraph.Iso.sumComm) a✝ = a✝.swap
• SimpleGraph.Iso.sumCongr_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Sum
  {V : Type u_3} {V' : Type u_4} {W : Type u_5} {W' : Type u_6} {G : SimpleGraph V} {H : SimpleGraph W} {G' : SimpleGraph V'} {H' : SimpleGraph W'} (f : G ≃g G') (g : H ≃g H') (a✝ : V' ⊕ W') : (RelIso.symm (f.sumCongr g)) a✝ = Sum.map (⇑f.symm) (⇑g.symm) a✝
• SimpleGraph.Iso.sumAssoc_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Sum
  {U : Type u_1} {V : Type u_3} {W : Type u_5} {G : SimpleGraph V} {H : SimpleGraph W} {I : SimpleGraph U} (a✝ : V ⊕ W ⊕ U) : (RelIso.symm SimpleGraph.Iso.sumAssoc) a✝ = Sum.elim (Sum.inl ∘ Sum.inl) (Sum.elim (Sum.inl ∘ Sum.inr) Sum.inr) a✝
• SimpleGraph.boxProdComm_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Prod
  {α : Type u_1} {β : Type u_2} (G : SimpleGraph α) (H : SimpleGraph β) (a✝ : β × α) : (RelIso.symm (G.boxProdComm H)) a✝ = a✝.swap
• SimpleGraph.boxProdAssoc_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (G : SimpleGraph α) (H : SimpleGraph β) (I : SimpleGraph γ) (p : α × β × γ) : (RelIso.symm (G.boxProdAssoc H I)) p = ((p.1, p.2.1), p.2.2)
• SimpleGraph.Iso.sumBoxProdDistrib_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Prod
  {V₁ : Type u_5} {V₂ : Type u_6} {W : Type u_7} (G₁ : SimpleGraph V₁) (G₂ : SimpleGraph V₂) (H : SimpleGraph W) (s : V₁ × W ⊕ V₂ × W) : (RelIso.symm (SimpleGraph.Iso.sumBoxProdDistrib G₁ G₂ H)) s = Sum.elim (Prod.map Sum.inl id) (Prod.map Sum.inr id) s
• SimpleGraph.Iso.boxProdSumDistrib_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.Prod
  {V : Type u_4} {W₁ : Type u_8} {W₂ : Type u_9} (G : SimpleGraph V) (H₁ : SimpleGraph W₁) (H₂ : SimpleGraph W₂) (a✝ : V × W₁ ⊕ V × W₂) : (RelIso.symm (SimpleGraph.Iso.boxProdSumDistrib G H₁ H₂)) a✝ = ((Equiv.sumProdDistrib W₁ W₂ V).symm (Sum.map Prod.swap Prod.swap a✝)).swap
• Matrix.IsAdjMatrix.toGraphReindexIso_symm_apply 📋 Mathlib.Combinatorics.SimpleGraph.AdjMatrix
  {α : Type u_1} {V : Type u_2} {W : Type u_3} (A : Matrix V V α) [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (f : V ≃ W) (a✝ : V) : (RelIso.symm (Matrix.IsAdjMatrix.toGraphReindexIso A h f)) a✝ = f a✝
• YoungDiagram.transposeOrderIso_symm_apply 📋 Mathlib.Combinatorics.Young.YoungDiagram
  (μ : YoungDiagram) : (RelIso.symm YoungDiagram.transposeOrderIso) μ = μ.transpose
• Concept.swapEquiv_symm_apply 📋 Mathlib.Order.Concept
  {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (a✝ : Concept β α (Function.swap r)) : (RelIso.symm Concept.swapEquiv) a✝ = (⇑OrderDual.toDual ∘ Concept.swap) a✝
• Subrepresentation.subrepresentationSubmoduleOrderIso_symm_apply 📋 Mathlib.RepresentationTheory.Subrepresentation
  {A : Type u_1} {G : Type u_2} {W : Type u_3} [CommSemiring A] [Monoid G] [AddCommMonoid W] [Module A W] {ρ : Representation A G W} (N : Submodule (MonoidAlgebra A G) ρ.asModule) : (RelIso.symm Subrepresentation.subrepresentationSubmoduleOrderIso) N = Subrepresentation.ofSubmodule' N
• Subrepresentation.submoduleSubrepresentationOrderIso_symm_apply 📋 Mathlib.RepresentationTheory.Subrepresentation
  {A : Type u_1} {G : Type u_2} {M : Type u_4} [CommSemiring A] [Monoid G] [AddCommMonoid M] [Module (MonoidAlgebra A G) M] (σ : Subrepresentation (Representation.ofModule M)) : (RelIso.symm Subrepresentation.submoduleSubrepresentationOrderIso) σ = σ.asSubmodule'
• OrderIso.setIsotypicComponents_symm_apply 📋 Mathlib.RingTheory.SimpleModule.Isotypic
  {R : Type u_2} {M : Type u} [Ring R] [AddCommGroup M] [Module R M] [IsSemisimpleModule R M] (m : ↥(fullyInvariantSubmodule R M)) : (RelIso.symm OrderIso.setIsotypicComponents) m = {c | ↑c ≤ ↑m}
• Ordinal.toZFSetIso_symm_apply 📋 Mathlib.SetTheory.ZFC.Ordinal
  (x : { x // x.IsOrdinal }) : (RelIso.symm Ordinal.toZFSetIso) x = (↑x).rank

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