Loogle!

Result

Found 132 declarations mentioning Equiv.toFun.

• Equiv.toFun 📋 Mathlib.Logic.Equiv.Defs
  {α : Sort u_1} {β : Sort u_2} (self : α ≃ β) : α → β
• Equiv.left_inv 📋 Mathlib.Logic.Equiv.Defs
  {α : Sort u_1} {β : Sort u_2} (self : α ≃ β) : Function.LeftInverse self.invFun self.toFun
• Equiv.right_inv 📋 Mathlib.Logic.Equiv.Defs
  {α : Sort u_1} {β : Sort u_2} (self : α ≃ β) : Function.RightInverse self.invFun self.toFun
• Equiv.toFun_as_coe 📋 Mathlib.Logic.Equiv.Defs
  {α : Sort u} {β : Sort v} (e : α ≃ β) : e.toFun = ⇑e
• Equiv.symmEquiv_apply_symm_apply 📋 Mathlib.Logic.Equiv.Defs
  (α : Sort u_1) (β : Sort u_2) (e : α ≃ β) (a✝ : α) : ((Equiv.symmEquiv α β) e).symm a✝ = e.toFun a✝
• Equiv.symmEquiv_symm_apply_symm_apply 📋 Mathlib.Logic.Equiv.Defs
  (α : Sort u_1) (β : Sort u_2) (e : β ≃ α) (a✝ : β) : ((Equiv.symmEquiv α β).symm e).symm a✝ = e.toFun a✝
• AddEquiv.toFun_eq_coe 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : f.toFun = ⇑f
• MulEquiv.toFun_eq_coe 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : f.toFun = ⇑f
• AddEquiv.mk 📋 Mathlib.Algebra.Group.Equiv.Defs
  {A : Type u_9} {B : Type u_10} [Add A] [Add B] (toEquiv : A ≃ B) (map_add' : ∀ (x y : A), toEquiv.toFun (x + y) = toEquiv.toFun x + toEquiv.toFun y) : A ≃+ B
• MulEquiv.mk 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_9} {N : Type u_10} [Mul M] [Mul N] (toEquiv : M ≃ N) (map_mul' : ∀ (x y : M), toEquiv.toFun (x * y) = toEquiv.toFun x * toEquiv.toFun y) : M ≃* N
• AddEquiv.map_add' 📋 Mathlib.Algebra.Group.Equiv.Defs
  {A : Type u_9} {B : Type u_10} [Add A] [Add B] (self : A ≃+ B) (x y : A) : self.toFun (x + y) = self.toFun x + self.toFun y
• MulEquiv.map_mul' 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_9} {N : Type u_10} [Mul M] [Mul N] (self : M ≃* N) (x y : M) : self.toFun (x * y) = self.toFun x * self.toFun y
• AddEquiv.symm_mk 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃ N) (h : ∀ (x y : M), f.toFun (x + y) = f.toFun x + f.toFun y) : { toEquiv := f, map_add' := h }.symm = { toEquiv := f.symm, map_add' := ⋯ }
• MulEquiv.symm_mk 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃ N) (h : ∀ (x y : M), f.toFun (x * y) = f.toFun x * f.toFun y) : { toEquiv := f, map_mul' := h }.symm = { toEquiv := f.symm, map_mul' := ⋯ }
• AddEquiv.mk_coe 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) (e' : N → M) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : M), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_add' := h₃ } = e
• MulEquiv.mk_coe 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) (e' : N → M) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : M), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃ } = e
• AddEquiv.mk_coe' 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) (f : N → M) (h₁ : Function.LeftInverse (⇑e) f) (h₂ : Function.RightInverse (⇑e) f) (h₃ : ∀ (x y : N), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂, map_add' := h₃ } = e.symm
• MulEquiv.mk_coe' 📋 Mathlib.Algebra.Group.Equiv.Defs
  {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) (f : N → M) (h₁ : Function.LeftInverse (⇑e) f) (h₂ : Function.RightInverse (⇑e) f) (h₃ : ∀ (x y : N), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂, map_mul' := h₃ } = e.symm
• OrderIso.toFun_eq_coe 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [LE α] [LE β] {f : α ≃o β} : f.toFun = ⇑f
• AddConstEquiv.mk 📋 Mathlib.Algebra.AddConstMap.Equiv
  {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (toEquiv : G ≃ H) (map_add_const' : ∀ (x : G), toEquiv.toFun (x + a) = toEquiv.toFun x + b) : AddConstEquiv G H a b
• AddConstEquiv.map_add_const' 📋 Mathlib.Algebra.AddConstMap.Equiv
  {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (self : AddConstEquiv G H a b) (x : G) : self.toFun (x + a) = self.toFun x + b
• OrderAddMonoidIso.mk 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [Add α] [Add β] (toAddEquiv : α ≃+ β) (map_le_map_iff' : ∀ {a b : α}, toAddEquiv.toFun a ≤ toAddEquiv.toFun b ↔ a ≤ b) : α ≃+o β
• OrderMonoidIso.mk 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [Mul α] [Mul β] (toMulEquiv : α ≃* β) (map_le_map_iff' : ∀ {a b : α}, toMulEquiv.toFun a ≤ toMulEquiv.toFun b ↔ a ≤ b) : α ≃*o β
• OrderAddMonoidIso.toFun_eq_coe 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [Add α] [Add β] (f : α ≃+o β) : f.toFun = ⇑f
• OrderMonoidIso.toFun_eq_coe 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [Mul α] [Mul β] (f : α ≃*o β) : f.toFun = ⇑f
• OrderAddMonoidIso.map_le_map_iff' 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [Add α] [Add β] (self : α ≃+o β) {a b : α} : self.toFun a ≤ self.toFun b ↔ a ≤ b
• OrderMonoidIso.map_le_map_iff' 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [Mul α] [Mul β] (self : α ≃*o β) {a b : α} : self.toFun a ≤ self.toFun b ↔ a ≤ b
• OrderAddMonoidIso.coe_mk 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [Add α] [Add β] (f : α ≃+ β) (h : ∀ {a b : α}, f.toFun a ≤ f.toFun b ↔ a ≤ b) : ⇑{ toAddEquiv := f, map_le_map_iff' := h } = ⇑f
• OrderMonoidIso.coe_mk 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [Mul α] [Mul β] (f : α ≃* β) (h : ∀ {a b : α}, f.toFun a ≤ f.toFun b ↔ a ≤ b) : ⇑{ toMulEquiv := f, map_le_map_iff' := h } = ⇑f
• OrderAddMonoidIso.mk_coe 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [Add α] [Add β] (f : α ≃+o β) (h : ∀ {a b : α}, (↑f).toFun a ≤ (↑f).toFun b ↔ a ≤ b) : { toAddEquiv := ↑f, map_le_map_iff' := h } = f
• OrderMonoidIso.mk_coe 📋 Mathlib.Algebra.Order.Hom.Monoid
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [Mul α] [Mul β] (f : α ≃*o β) (h : ∀ {a b : α}, (↑f).toFun a ≤ (↑f).toFun b ↔ a ≤ b) : { toMulEquiv := ↑f, map_le_map_iff' := h } = f
• RingEquiv.map_add' 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [Mul R] [Mul S] [Add R] [Add S] (self : R ≃+* S) (x y : R) : self.toFun (x + y) = self.toFun x + self.toFun y
• RingEquiv.map_mul' 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [Mul R] [Mul S] [Add R] [Add S] (self : R ≃+* S) (x y : R) : self.toFun (x * y) = self.toFun x * self.toFun y
• RingEquiv.mk 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [Mul R] [Mul S] [Add R] [Add S] (toEquiv : R ≃ S) (map_mul' : ∀ (x y : R), toEquiv.toFun (x * y) = toEquiv.toFun x * toEquiv.toFun y) (map_add' : ∀ (x y : R), toEquiv.toFun (x + y) = toEquiv.toFun x + toEquiv.toFun y) : R ≃+* S
• RingEquiv.coe_mk 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃ S) (h₃ : ∀ (x y : R), e.toFun (x * y) = e.toFun x * e.toFun y) (h₄ : ∀ (x y : R), e.toFun (x + y) = e.toFun x + e.toFun y) : ⇑{ toEquiv := e, map_mul' := h₃, map_add' := h₄ } = ⇑e
• RingEquiv.symm_mk 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃ S) (h₁ : ∀ (x y : R), e.toFun (x * y) = e.toFun x * e.toFun y) (h₂ : ∀ (x y : R), e.toFun (x + y) = e.toFun x + e.toFun y) : { toEquiv := e, map_mul' := h₁, map_add' := h₂ }.symm = { toEquiv := e.symm, map_mul' := ⋯, map_add' := ⋯ }
• RingEquiv.mk_coe 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (e' : S → R) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : R), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : R), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ } = e
• RingEquiv.mk_coe' 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (f : S → R) (h₁ : Function.LeftInverse (⇑e) f) (h₂ : Function.RightInverse (⇑e) f) (h₃ : ∀ (x y : S), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : S), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ } = e.symm
• OrderRingIso.mk 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_6} {β : Type u_7} [Mul α] [Add α] [Mul β] [Add β] [LE α] [LE β] (toRingEquiv : α ≃+* β) (map_le_map_iff' : ∀ {a b : α}, toRingEquiv.toFun a ≤ toRingEquiv.toFun b ↔ a ≤ b) : α ≃+*o β
• OrderRingIso.map_le_map_iff' 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_6} {β : Type u_7} [Mul α] [Add α] [Mul β] [Add β] [LE α] [LE β] (self : α ≃+*o β) {a b : α} : self.toFun a ≤ self.toFun b ↔ a ≤ b
• OrderRingIso.toFun_eq_coe 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : f.toFun = ⇑f
• OrderRingIso.mk_coe 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) (h : ∀ {a b : α}, e.toFun a ≤ e.toFun b ↔ a ≤ b) : { toRingEquiv := e.toRingEquiv, map_le_map_iff' := h } = e
• OrderRingIso.coe_mk 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+* β) (h : ∀ {a b : α}, e.toFun a ≤ e.toFun b ↔ a ≤ b) : ⇑{ toRingEquiv := e, map_le_map_iff' := h } = ⇑e
• LinearEquiv.piUnique_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {α : Type u_9} [Unique α] (R : Type u_10) [Semiring R] (f : α → Type u_11) [(x : α) → AddCommMonoid (f x)] [(x : α) → Module R (f x)] : ⇑(LinearEquiv.piUnique R f) = (Equiv.piUnique f).toFun
• LinearEquiv.congrLeft_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  (M : Type u_5) {M₂ : Type u_7} {M₃ : Type u_8} [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] {R : Type u_9} (S : Type u_10) [Semiring R] [Semiring S] [Module R M₂] [Module R M₃] [Module R M] [Module S M] [SMulCommClass R S M] (e : M₂ ≃ₗ[R] M₃) (a✝ : M₂ →ₗ[R] M) : (LinearEquiv.congrLeft M S e) a✝ = (e.arrowCongrAddEquiv (LinearEquiv.refl R M)).toFun a✝
• LinearEquiv.domMulActCongrRight_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {S : Type u_4} {R₁ : Type u_9} {R₁' : Type u_11} {R₂' : Type u_12} {M₁ : Type u_13} {M₁' : Type u_15} {M₂' : Type u_16} [Semiring R₁] [Semiring R₁'] [Semiring R₂'] [AddCommMonoid M₁] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁ M₁] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [Semiring S] [Module S M₁] [SMulCommClass R₁ S M₁] [RingHomCompTriple σ₁₂' σ₂'₁' σ₁₁'] (e₂ : M₁' ≃ₛₗ[σ₁'₂'] M₂') (a✝ : M₁ →ₛₗ[σ₁₁'] M₁') : e₂.domMulActCongrRight a✝ = ((LinearEquiv.refl R₁ M₁).arrowCongrAddEquiv e₂).toFun a✝
• AlgEquiv.map_add' 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (self : A ≃ₐ[R] B) (x y : A) : self.toFun (x + y) = self.toFun x + self.toFun y
• AlgEquiv.map_mul' 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (self : A ≃ₐ[R] B) (x y : A) : self.toFun (x * y) = self.toFun x * self.toFun y
• AlgEquiv.commutes' 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (self : A ≃ₐ[R] B) (r : R) : self.toFun ((algebraMap R A) r) = (algebraMap R B) r
• AlgEquiv.mk 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (toEquiv : A ≃ B) (map_mul' : ∀ (x y : A), toEquiv.toFun (x * y) = toEquiv.toFun x * toEquiv.toFun y) (map_add' : ∀ (x y : A), toEquiv.toFun (x + y) = toEquiv.toFun x + toEquiv.toFun y) (commutes' : ∀ (r : R), toEquiv.toFun ((algebraMap R A) r) = (algebraMap R B) r) : A ≃ₐ[R] B
• AlgEquiv.coe_mk 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂] {toEquiv : A₁ ≃ A₂} {map_mul : ∀ (x y : A₁), toEquiv.toFun (x * y) = toEquiv.toFun x * toEquiv.toFun y} {map_add : ∀ (x y : A₁), toEquiv.toFun (x + y) = toEquiv.toFun x + toEquiv.toFun y} {commutes : ∀ (r : R), toEquiv.toFun ((algebraMap R A₁) r) = (algebraMap R A₂) r} : ⇑{ toEquiv := toEquiv, map_mul' := map_mul, map_add' := map_add, commutes' := commutes } = ⇑toEquiv
• AlgEquiv.symm_mk 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ A₂) (h₁ : ∀ (x y : A₁), e.toFun (x * y) = e.toFun x * e.toFun y) (h₂ : ∀ (x y : A₁), e.toFun (x + y) = e.toFun x + e.toFun y) (h₃ : ∀ (r : R), e.toFun ((algebraMap R A₁) r) = (algebraMap R A₂) r) : { toEquiv := e, map_mul' := h₁, map_add' := h₂, commutes' := h₃ }.symm = { toEquiv := e.symm, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
• AlgEquiv.mk_coe 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) (e' : A₂ → A₁) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : A₁), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A₁), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (r : R), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun ((algebraMap R A₁) r) = (algebraMap R A₂) r) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, commutes' := h₅ } = e
• AlgEquiv.mk_coe' 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) (f : A₂ → A₁) (h₁ : Function.LeftInverse (⇑e) f) (h₂ : Function.RightInverse (⇑e) f) (h₃ : ∀ (x y : A₂), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A₂), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (r : R), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun ((algebraMap R A₂) r) = (algebraMap R A₁) r) : { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, commutes' := h₅ } = e.symm
• finTwoArrowEquiv_apply 📋 Mathlib.Logic.Equiv.Fin.Basic
  (α : Type u_1) : ⇑(finTwoArrowEquiv α) = (piFinTwoEquiv fun x => α).toFun
• LinearEquiv.prodAssoc_apply 📋 Mathlib.LinearAlgebra.Prod
  (R : Type u_3) (M₁ : Type u_4) (M₂ : Type u_5) (M₃ : Type u_6) [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₁] [Module R M₂] [Module R M₃] (a✝ : (M₁ × M₂) × M₃) : (LinearEquiv.prodAssoc R M₁ M₂ M₃) a✝ = AddEquiv.prodAssoc.toFun a✝
• Fin.consLinearEquiv_apply 📋 Mathlib.LinearAlgebra.Pi
  (R : Type u) {n : ℕ} (M : Fin n.succ → Type u_1) [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [(i : Fin n.succ) → Module R (M i)] (a✝ : M 0 × ((i : Fin n) → M i.succ)) (i : Fin (n + 1)) : (Fin.consLinearEquiv R M) a✝ i = (Fin.consEquiv M).toFun a✝ i
• DFinsupp.equivCongrLeft_apply 📋 Mathlib.Data.DFinsupp.Defs
  {ι : Type u} {β : ι → Type v} {κ : Type u_1} [(i : ι) → Zero (β i)] (h : ι ≃ κ) (f : Π₀ (i : ι), β i) : (DFinsupp.equivCongrLeft h) f = DFinsupp.comapDomain' ⇑h.symm ⋯ f
• sigmaFinsuppLequivDFinsupp_apply 📋 Mathlib.Data.Finsupp.ToDFinsupp
  {ι : Type u_1} (R : Type u_2) {η : ι → Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] [Module R N] (a✝ : (i : ι) × η i →₀ N) : (sigmaFinsuppLequivDFinsupp R) a✝ = sigmaFinsuppAddEquivDFinsupp.toFun a✝
• Finsupp.sumFinsuppLEquivProdFinsupp_apply 📋 Mathlib.LinearAlgebra.Finsupp.SumProd
  {M : Type u_2} (R : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] {α : Type u_4} {β : Type u_5} (a✝ : α ⊕ β →₀ M) : (Finsupp.sumFinsuppLEquivProdFinsupp R) a✝ = Finsupp.sumFinsuppAddEquivProdFinsupp.toFun a✝
• Matrix.transposeLinearEquiv_apply 📋 Mathlib.Data.Matrix.Basic
  (m : Type u_2) (n : Type u_3) (R : Type u_4) (α : Type u_8) [Semiring R] [AddCommMonoid α] [Module R α] (a✝ : Matrix m n α) : (Matrix.transposeLinearEquiv m n R α) a✝ = (Matrix.transposeAddEquiv m n α).toFun a✝
• Matrix.piLinearEquiv_apply 📋 Mathlib.Data.Matrix.Basic
  {m : Type u_2} {n : Type u_3} {ι : Type u_11} {β : ι → Type u_12} (R : Type u_13) [Semiring R] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → Module R (β i)] (a✝ : Matrix m n ((i : ι) → β i)) (i : ι) : (Matrix.piLinearEquiv R) a✝ i = Matrix.piAddEquiv.toFun a✝ i
• DirectSum.lequivProdDirectSum_apply 📋 Mathlib.Algebra.DirectSum.Module
  (R : Type u) [Semiring R] {ι : Type v} {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] [(i : Option ι) → Module R (α i)] (a✝ : DirectSum (Option ι) fun i => α i) : (DirectSum.lequivProdDirectSum R) a✝ = DirectSum.addEquivProdDirectSum.toFun a✝
• StarRingEquiv.mk 📋 Mathlib.Algebra.Star.StarRingHom
  {A : Type u_1} {B : Type u_2} [Add A] [Add B] [Mul A] [Mul B] [Star A] [Star B] (toRingEquiv : A ≃+* B) (map_star' : ∀ (a : A), toRingEquiv.toFun (star a) = star (toRingEquiv.toFun a)) : A ≃⋆+* B
• StarRingEquiv.map_star' 📋 Mathlib.Algebra.Star.StarRingHom
  {A : Type u_1} {B : Type u_2} [Add A] [Add B] [Mul A] [Mul B] [Star A] [Star B] (self : A ≃⋆+* B) (a : A) : self.toFun (star a) = star (self.toFun a)
• StarRingEquiv.coe_mk 📋 Mathlib.Algebra.Star.StarRingHom
  {A : Type u_1} {B : Type u_2} [Add A] [Add B] [Mul A] [Mul B] [Star A] [Star B] (e : A ≃+* B) (h₁ : ∀ (a : A), e.toFun (star a) = star (e.toFun a)) : ⇑{ toRingEquiv := e, map_star' := h₁ } = ⇑e
• StarRingEquiv.symm_mk 📋 Mathlib.Algebra.Star.StarRingHom
  {A : Type u_1} {B : Type u_2} [Add A] [Add B] [Mul A] [Mul B] [Star A] [Star B] (e : A ≃+* B) (h₁ : ∀ (a : A), e.toFun (star a) = star (e.toFun a)) : { toRingEquiv := e, map_star' := h₁ }.symm = { toRingEquiv := e.symm, map_star' := ⋯ }
• StarRingEquiv.mk_coe 📋 Mathlib.Algebra.Star.StarRingHom
  {A : Type u_1} {B : Type u_2} [Add A] [Add B] [Mul A] [Mul B] [Star A] [Star B] (e : A ≃⋆+* B) (e' : B → A) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (a : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) = star ({ toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅ } = e
• StarAlgEquiv.mk 📋 Mathlib.Algebra.Star.StarAlgHom
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (toStarRingEquiv : A ≃⋆+* B) (map_smul' : ∀ (r : R) (a : A), toStarRingEquiv.toFun (r • a) = r • toStarRingEquiv.toFun a) : A ≃⋆ₐ[R] B
• StarAlgEquiv.map_smul' 📋 Mathlib.Algebra.Star.StarAlgHom
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (self : A ≃⋆ₐ[R] B) (r : R) (a : A) : self.toFun (r • a) = r • self.toFun a
• StarAlgEquiv.coe_mk 📋 Mathlib.Algebra.Star.StarAlgHom
  {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆+* B) (h : ∀ (r : R) (a : A), e.toFun (r • a) = r • e.toFun a) : ⇑{ toStarRingEquiv := e, map_smul' := h } = ⇑e
• StarAlgEquiv.symm_mk 📋 Mathlib.Algebra.Star.StarAlgHom
  {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆+* B) (h₁ : ∀ (r : R) (a : A), e.toFun (r • a) = r • e.toFun a) : { toStarRingEquiv := e, map_smul' := h₁ }.symm = { toStarRingEquiv := e.symm, map_smul' := ⋯ }
• StarAlgEquiv.mk_coe 📋 Mathlib.Algebra.Star.StarAlgHom
  {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (e' : B → A) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (a : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) = star ({ toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) (h₆ : ∀ (r : R) (a : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅ }.toFun (r • a) = r • { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅ }.toFun a) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅, map_smul' := h₆ } = e
• Matrix.conjTransposeLinearEquiv_apply 📋 Mathlib.LinearAlgebra.Matrix.ConjTranspose
  (m : Type u_2) (n : Type u_3) (R : Type u_5) (α : Type v) [CommSemiring R] [StarRing R] [AddCommMonoid α] [StarAddMonoid α] [Module R α] [StarModule R α] (a✝ : Matrix m n α) : (Matrix.conjTransposeLinearEquiv m n R α) a✝ = (Matrix.conjTransposeAddEquiv m n α).toFun a✝
• SemimoduleCat.homLinearEquiv_apply 📋 Mathlib.Algebra.Category.ModuleCat.Semi
  {R : Type u} [Semiring R] {M N : SemimoduleCat R} {S : Type u_1} [Semiring S] [Module S ↑N] [SMulCommClass R S ↑N] (a✝ : M ⟶ N) : SemimoduleCat.homLinearEquiv a✝ = SemimoduleCat.homAddEquiv.toFun a✝
• ModuleCat.homLinearEquiv_apply 📋 Mathlib.Algebra.Category.ModuleCat.Basic
  {R : Type u} [Ring R] {M N : ModuleCat R} {S : Type u_1} [Semiring S] [Module S ↑N] [SMulCommClass R S ↑N] (a✝ : M ⟶ N) : ModuleCat.homLinearEquiv a✝ = ModuleCat.homAddEquiv.toFun a✝
• MultilinearMap.domDomCongrLinearEquiv_apply 📋 Mathlib.LinearAlgebra.Multilinear.Basic
  (R : Type uR) (S : Type uS) (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [AddCommMonoid M₃] [Module S M₃] [Module R M₃] [SMulCommClass R S M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂) (a✝ : MultilinearMap R (fun x => M₂) M₃) : (MultilinearMap.domDomCongrLinearEquiv R S M₂ M₃ σ) a✝ = (MultilinearMap.domDomCongrEquiv σ).toFun a✝
• BialgEquiv.ofAlgEquiv_apply 📋 Mathlib.RingTheory.Bialgebra.Equiv
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Bialgebra R A] [Bialgebra R B] (f : A ≃ₐ[R] B) (counit_comp : (Bialgebra.counitAlgHom R B).comp ↑f = Bialgebra.counitAlgHom R A) (map_comp_comul : (Algebra.TensorProduct.map ↑f ↑f).comp (Bialgebra.comulAlgHom R A) = (Bialgebra.comulAlgHom R B).comp ↑f) (a✝ : A) : (BialgEquiv.ofAlgEquiv f counit_comp map_comp_comul) a✝ = f.toFun a✝
• CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_mk 📋 Mathlib.CategoryTheory.Elements
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v)) {X : C} (f : CategoryTheory.yoneda.obj X ⟶ F) : (CategoryTheory.CategoryOfElements.fromCostructuredArrow F).obj (Opposite.op (CategoryTheory.CostructuredArrow.mk f)) = ⟨Opposite.op X, CategoryTheory.yonedaEquiv.toFun f⟩
• CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_snd 📋 Mathlib.CategoryTheory.Elements
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v)) (X : (CategoryTheory.CostructuredArrow CategoryTheory.yoneda F)ᵒᵖ) : ((CategoryTheory.CategoryOfElements.fromCostructuredArrow F).obj X).snd = CategoryTheory.yonedaEquiv.toFun (Opposite.unop X).hom
• CategoryTheory.CategoryOfElements.fromCostructuredArrow_map_coe 📋 Mathlib.CategoryTheory.Elements
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v)) {X Y : (CategoryTheory.CostructuredArrow CategoryTheory.yoneda F)ᵒᵖ} (f : X ⟶ Y) : ↑((CategoryTheory.CategoryOfElements.fromCostructuredArrow F).map f) = f.unop.left.op
• Homeomorph.continuous_toFun 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] (self : X ≃ₜ Y) : Continuous self.toFun
• Homeomorph.mk 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] (toEquiv : X ≃ Y) (continuous_toFun : Continuous toEquiv.toFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) (continuous_invFun : Continuous toEquiv.invFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) : X ≃ₜ Y
• Homeomorph.homeomorph_mk_coe 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (a : X ≃ Y) (b : Continuous a.toFun) (c : Continuous a.invFun) : ⇑{ toEquiv := a, continuous_toFun := b, continuous_invFun := c } = ⇑a
• Homeomorph.homeomorph_mk_coe_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (a : X ≃ Y) (b : Continuous a.toFun) (c : Continuous a.invFun) : ⇑{ toEquiv := a, continuous_toFun := b, continuous_invFun := c }.symm = ⇑a.symm
• UniformEquiv.uniformContinuous_toFun 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (self : α ≃ᵤ β) : UniformContinuous self.toFun
• UniformEquiv.mk 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (toEquiv : α ≃ β) (uniformContinuous_toFun : UniformContinuous toEquiv.toFun) (uniformContinuous_invFun : UniformContinuous toEquiv.invFun) : α ≃ᵤ β
• UniformEquiv.uniformEquiv_mk_coe 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun) : ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c } = ⇑a
• UniformEquiv.uniformEquiv_mk_coe_symm 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun) : ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c }.symm = ⇑a.symm
• ContinuousAddEquiv.continuous_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {G : Type u} [TopologicalSpace G] {H : Type v} [TopologicalSpace H] [Add G] [Add H] (self : G ≃ₜ+ H) : Continuous self.toFun
• ContinuousMulEquiv.continuous_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {G : Type u} [TopologicalSpace G] {H : Type v} [TopologicalSpace H] [Mul G] [Mul H] (self : G ≃ₜ* H) : Continuous self.toFun
• ContinuousAddEquiv.mk 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {G : Type u} [TopologicalSpace G] {H : Type v} [TopologicalSpace H] [Add G] [Add H] (toAddEquiv : G ≃+ H) (continuous_toFun : Continuous toAddEquiv.toFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) (continuous_invFun : Continuous toAddEquiv.invFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) : G ≃ₜ+ H
• ContinuousMulEquiv.mk 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {G : Type u} [TopologicalSpace G] {H : Type v} [TopologicalSpace H] [Mul G] [Mul H] (toMulEquiv : G ≃* H) (continuous_toFun : Continuous toMulEquiv.toFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) (continuous_invFun : Continuous toMulEquiv.invFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) : G ≃ₜ* H
• ContinuousAddEquiv.coe_mk 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {M : Type u_1} {N : Type u_2} [TopologicalSpace M] [TopologicalSpace N] [Add M] [Add N] (f : M ≃+ N) (hf1 : Continuous f.toFun) (hf2 : Continuous f.invFun) : ⇑{ toAddEquiv := f, continuous_toFun := hf1, continuous_invFun := hf2 } = ⇑f
• ContinuousMulEquiv.coe_mk 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {M : Type u_1} {N : Type u_2} [TopologicalSpace M] [TopologicalSpace N] [Mul M] [Mul N] (f : M ≃* N) (hf1 : Continuous f.toFun) (hf2 : Continuous f.invFun) : ⇑{ toMulEquiv := f, continuous_toFun := hf1, continuous_invFun := hf2 } = ⇑f
• CategoryTheory.Abelian.Ext.homLinearEquiv_apply 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear
  {R : Type t} [Ring R] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Linear R C] [CategoryTheory.HasExt C] {X Y : C} {n : ℕ} [HasDerivedCategory C] (a✝ : CategoryTheory.Abelian.Ext X Y n) : CategoryTheory.Abelian.Ext.homLinearEquiv a✝ = CategoryTheory.Abelian.Ext.homAddEquiv.toFun a✝
• ContinuousLinearMap.prodₗ_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_4} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R M₃] (S : Type u_5) [Semiring S] [Module S M₂] [ContinuousAdd M₂] [SMulCommClass R S M₂] [ContinuousConstSMul S M₂] [Module S M₃] [ContinuousAdd M₃] [SMulCommClass R S M₃] [ContinuousConstSMul S M₃] (a✝ : (M →L[R] M₂) × (M →L[R] M₃)) : (ContinuousLinearMap.prodₗ S) a✝ = ContinuousLinearMap.prodEquiv.toFun a✝
• ContinuousLinearEquiv.arrowCongrEquivₛₗ_apply 📋 Mathlib.Topology.Algebra.Module.Equiv
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] {R₄ : Type u_8} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [SMulCommClass R₃ R₃ M₃] [SMulCommClass R₄ R₄ M₄] [ContinuousAdd M₃] [ContinuousConstSMul R₃ M₃] [ContinuousAdd M₄] [ContinuousConstSMul R₄ M₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) (a✝ : M₁ →SL[σ₁₄] M₄) : (e₁₂.arrowCongrEquivₛₗ e₄₃) a✝ = (e₁₂.arrowCongrEquiv e₄₃).toFun a✝
• Representation.ofMulActionSelfAsModuleEquiv_apply 📋 Mathlib.RepresentationTheory.Basic
  {k : Type u_1} {G : Type u_2} [CommSemiring k] [Group G] (a✝ : (Representation.ofMulAction k G G).asModule) : Representation.ofMulActionSelfAsModuleEquiv a✝ = (Representation.ofMulAction k G G).asModuleEquiv.toAddEquiv.toFun a✝
• IsometryEquiv.mk 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (toEquiv : α ≃ β) (isometry_toFun : Isometry toEquiv.toFun) : α ≃ᵢ β
• IsometryEquiv.isometry_toFun 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (self : α ≃ᵢ β) : Isometry self.toFun
• IsometryEquiv.coe_mk 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (e : α ≃ β) (h : Isometry e.toFun) : ⇑{ toEquiv := e, isometry_toFun := h } = ⇑e
• DilationEquiv.mk 📋 Mathlib.Topology.MetricSpace.DilationEquiv
  {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (toEquiv : X ≃ Y) (edist_eq' : ∃ r, r ≠ 0 ∧ ∀ (x y : X), edist (toEquiv.toFun x) (toEquiv.toFun y) = ↑r * edist x y) : X ≃ᵈ Y
• DilationEquiv.edist_eq' 📋 Mathlib.Topology.MetricSpace.DilationEquiv
  {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (self : X ≃ᵈ Y) : ∃ r, r ≠ 0 ∧ ∀ (x y : X), edist (self.toFun x) (self.toFun y) = ↑r * edist x y
• StarMulEquiv.mk 📋 Mathlib.Algebra.Star.MonoidHom
  {A : Type u_6} {B : Type u_7} [Mul A] [Mul B] [Star A] [Star B] (toMulEquiv : A ≃* B) (map_star' : ∀ (a : A), toMulEquiv.toFun (star a) = star (toMulEquiv.toFun a)) : A ≃⋆* B
• StarMulEquiv.map_star' 📋 Mathlib.Algebra.Star.MonoidHom
  {A : Type u_6} {B : Type u_7} [Mul A] [Mul B] [Star A] [Star B] (self : A ≃⋆* B) (a : A) : self.toFun (star a) = star (self.toFun a)
• StarMulEquiv.coe_mk 📋 Mathlib.Algebra.Star.MonoidHom
  {A : Type u_2} {B : Type u_3} [Mul A] [Mul B] [Star A] [Star B] (e : A ≃* B) (h₁ : ∀ (a : A), e.toFun (star a) = star (e.toFun a)) : ⇑{ toMulEquiv := e, map_star' := h₁ } = ⇑e
• RCLike.realLinearIsometryEquiv_apply 📋 Mathlib.Analysis.RCLike.Basic
  {K : Type u_1} [RCLike K] (h : RCLike.I = 0) (a✝ : K) : (RCLike.realLinearIsometryEquiv h) a✝ = (RCLike.realRingEquiv h).toFun a✝
• ContinuousAlgEquiv.continuous_toFun 📋 Mathlib.Topology.Algebra.Algebra.Equiv
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [TopologicalSpace A] [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (self : A ≃A[R] B) : Continuous self.toFun
• ContinuousAlgEquiv.mk 📋 Mathlib.Topology.Algebra.Algebra.Equiv
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [TopologicalSpace A] [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (toAlgEquiv : A ≃ₐ[R] B) (continuous_toFun : Continuous toAlgEquiv.toFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) (continuous_invFun : Continuous toAlgEquiv.invFun := by first | fun_prop | eta_expand; dsimp; fun_prop | skip) : A ≃A[R] B
• ContinuousAlgEquiv.coe_mk 📋 Mathlib.Topology.Algebra.Algebra.Equiv
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [TopologicalSpace A] [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (e : A ≃ₐ[R] B) (he : Continuous e.toFun) (he' : Continuous e.invFun) : ⇑{ toAlgEquiv := e, continuous_toFun := he, continuous_invFun := he' } = ⇑e
• RCLike.complexLinearIsometryEquiv_apply 📋 Mathlib.Analysis.Complex.Basic
  {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) (a✝ : 𝕜) : (RCLike.complexLinearIsometryEquiv h) a✝ = (RCLike.complexRingEquiv h).toFun a✝
• ContinuousMultilinearMap.ofSubsingletonₗᵢ_apply 📋 Mathlib.Analysis.Normed.Module.Multilinear.Basic
  (𝕜 : Type u) {ι : Type v} (G : Type wG) {G' : Type wG'} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Fintype ι] [Subsingleton ι] (i : ι) (a✝ : G →L[𝕜] G') : (ContinuousMultilinearMap.ofSubsingletonₗᵢ 𝕜 G i) a✝ = (ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i).toFun a✝
• ContinuousMultilinearMap.prodL_apply 📋 Mathlib.Analysis.Normed.Module.Multilinear.Basic
  (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (G : Type wG) (G' : Type wG') [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Fintype ι] (a✝ : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G') : (ContinuousMultilinearMap.prodL 𝕜 E G G') a✝ = ContinuousMultilinearMap.prodEquiv.toFun a✝
• ContinuousAffineEquiv.continuous_toFun 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
  {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_4} {V₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂] (self : P₁ ≃ᴬ[k] P₂) : Continuous (↑self).toFun
• ContinuousAffineEquiv.mk 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
  {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_4} {V₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂] (toAffineEquiv : P₁ ≃ᵃ[k] P₂) (continuous_toFun : Continuous toAffineEquiv.toFun := by fun_prop) (continuous_invFun : Continuous toAffineEquiv.invFun := by fun_prop) : P₁ ≃ᴬ[k] P₂
• WithLp.linearEquiv_apply 📋 Mathlib.Analysis.Normed.Lp.WithLp
  (p : ENNReal) (K : Type u_1) (V : Type u_4) [Semiring K] [AddCommGroup V] [Module K V] (a✝ : WithLp p V) : (WithLp.linearEquiv p K V) a✝ = (WithLp.addEquiv p V).toFun a✝
• CategoryTheory.Endofunctor.Adjunction.Algebra.toCoalgebraOf_map_f 📋 Mathlib.CategoryTheory.Endofunctor.Algebra
  {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor C C} (adj : F ⊣ G) {X✝ Y✝ : CategoryTheory.Endofunctor.Algebra F} (f : X✝ ⟶ Y✝) : ((CategoryTheory.Endofunctor.Adjunction.Algebra.toCoalgebraOf adj).map f).f = f.f
• CategoryTheory.Endofunctor.Adjunction.algebraCoalgebraEquiv_functor_map_f 📋 Mathlib.CategoryTheory.Endofunctor.Algebra
  {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor C C} (adj : F ⊣ G) {X✝ Y✝ : CategoryTheory.Endofunctor.Algebra F} (f : X✝ ⟶ Y✝) : ((CategoryTheory.Endofunctor.Adjunction.algebraCoalgebraEquiv adj).functor.map f).f = f.f
• CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_apply 📋 Mathlib.CategoryTheory.Preadditive.HomOrthogonal
  {C : Type u} [CategoryTheory.Category.{v, u} C] {ι : Type u_1} {s : ι → C} [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteBiproducts C] {R : Type u_2} [Semiring R] [CategoryTheory.Linear R C] (o : CategoryTheory.HomOrthogonal s) {α β : Type} [Finite α] [Finite β] {f : α → ι} {g : β → ι} (a✝ : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)) (i : ι) : o.matrixDecompositionLinearEquiv a✝ i = o.matrixDecompositionAddEquiv.toFun a✝ i
• FirstOrder.Language.Equiv.map_rel' 📋 Mathlib.ModelTheory.Basic
  {L : FirstOrder.Language} {M : Type w} {N : Type w'} [L.Structure M] [L.Structure N] (self : L.Equiv M N) {n : ℕ} (r : L.Relations n) (x : Fin n → M) : FirstOrder.Language.Structure.RelMap r (self.toFun ∘ x) ↔ FirstOrder.Language.Structure.RelMap r x
• FirstOrder.Language.Equiv.map_fun' 📋 Mathlib.ModelTheory.Basic
  {L : FirstOrder.Language} {M : Type w} {N : Type w'} [L.Structure M] [L.Structure N] (self : L.Equiv M N) {n : ℕ} (f : L.Functions n) (x : Fin n → M) : self.toFun (FirstOrder.Language.Structure.funMap f x) = FirstOrder.Language.Structure.funMap f (self.toFun ∘ x)
• FirstOrder.Language.Equiv.mk 📋 Mathlib.ModelTheory.Basic
  {L : FirstOrder.Language} {M : Type w} {N : Type w'} [L.Structure M] [L.Structure N] (toEquiv : M ≃ N) (map_fun' : ∀ {n : ℕ} (f : L.Functions n) (x : Fin n → M), toEquiv.toFun (FirstOrder.Language.Structure.funMap f x) = FirstOrder.Language.Structure.funMap f (toEquiv.toFun ∘ x) := by intros; trivial) (map_rel' : ∀ {n : ℕ} (r : L.Relations n) (x : Fin n → M), FirstOrder.Language.Structure.RelMap r (toEquiv.toFun ∘ x) ↔ FirstOrder.Language.Structure.RelMap r x := by intros; trivial) : L.Equiv M N
• MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_apply 📋 Mathlib.MeasureTheory.Measure.Complex
  {α : Type u_1} {m : MeasurableSpace α} {R : Type u_2} [Semiring R] [Module R ℝ] [ContinuousConstSMul R ℝ] [ContinuousConstSMul R ℂ] (a✝ : MeasureTheory.ComplexMeasure α) : MeasureTheory.ComplexMeasure.equivSignedMeasureₗ a✝ = MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun a✝
• Matrix.uniqueLinearEquiv_apply 📋 Mathlib.LinearAlgebra.Matrix.Unique
  {m : Type u_1} {n : Type u_2} {A : Type u_3} {R : Type u_4} [Unique m] [Unique n] [Semiring R] [AddCommMonoid A] [Module R A] (a✝ : Matrix m n A) : Matrix.uniqueLinearEquiv a✝ = Matrix.uniqueAddEquiv.toFun a✝
• RingInvo.involution' 📋 Mathlib.RingTheory.RingInvo
  {R : Type u_2} [Semiring R] (self : RingInvo R) (x : R) : MulOpposite.unop (self.toFun (MulOpposite.unop (self.toFun x))) = x
• RingInvo.mk 📋 Mathlib.RingTheory.RingInvo
  {R : Type u_2} [Semiring R] (toRingEquiv : R ≃+* Rᵐᵒᵖ) (involution' : ∀ (x : R), MulOpposite.unop (toRingEquiv.toFun (MulOpposite.unop (toRingEquiv.toFun x))) = x) : RingInvo R
• Profinite.NobelingProof.GoodProducts.equiv_toFun_eq_eval 📋 Mathlib.Topology.Category.Profinite.Nobeling.Basic
  {I : Type u} (C : Set (I → Bool)) [LinearOrder I] : (Profinite.NobelingProof.GoodProducts.equiv_range C).toFun = Set.rangeFactorization (Profinite.NobelingProof.GoodProducts.eval C)
• Profinite.NobelingProof.GoodProducts.sum_equiv_comp_eval_eq_elim 📋 Mathlib.Topology.Category.Profinite.Nobeling.Successor
  {I : Type u} (C : Set (I → Bool)) [LinearOrder I] [WellFoundedLT I] {o : Ordinal.{u}} (hsC : Profinite.NobelingProof.contained C (Order.succ o)) (ho : o < Ordinal.type fun x1 x2 => x1 < x2) : Profinite.NobelingProof.GoodProducts.eval C ∘ (Profinite.NobelingProof.GoodProducts.sum_equiv C hsC ho).toFun = Sum.elim (fun l => Profinite.NobelingProof.Products.eval C ↑l) fun l => Profinite.NobelingProof.Products.eval C ↑l
• Profinite.NobelingProof.GoodProducts.range_equiv_factorization 📋 Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit
  {I : Type u} (C : Set (I → Bool)) [LinearOrder I] [WellFoundedLT I] {o : Ordinal.{u}} (ho : Order.IsSuccLimit o) (hsC : Profinite.NobelingProof.contained C o) : (fun p => ↑p) ∘ (Profinite.NobelingProof.GoodProducts.range_equiv C ho hsC).toFun = fun p => ↑p
• Profinite.NobelingProof.GoodProducts.smaller_factorization 📋 Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit
  {I : Type u} (C : Set (I → Bool)) [LinearOrder I] [WellFoundedLT I] (o : Ordinal.{u}) : (fun p => ↑p) ∘ (Profinite.NobelingProof.GoodProducts.range_equiv_smaller C o).toFun = ⇑(Profinite.NobelingProof.πs C o) ∘ fun p => ↑p
• GenLoop.genLoopGenLoopEquiv_apply_coe 📋 Mathlib.Topology.Homotopy.HomotopyGroup
  {N : Type u_1} {X : Type u_2} [TopologicalSpace X] {M : Type u_3} (x : X) (p : ↑(GenLoop M (↑(GenLoop N X x)) GenLoop.const)) : ↑((GenLoop.genLoopGenLoopEquiv x) p) = (GenLoop.uncurry x p).comp { toFun := Homeomorph.sumArrowHomeomorphProdArrow.toFun, continuous_toFun := ⋯ }

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