Loogle!

Result

Found 1348 declarations mentioning RingEquiv. Of these, only the first 200 are shown.

• RingEquiv 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_7) (S : Type u_8) [Mul R] [Mul S] [Add R] [Add S] : Type (max u_7 u_8)
• RingEquiv.refl 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [Mul R] [Add R] : R ≃+* R
• RingEquiv.instInhabited 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [Mul R] [Add R] : Inhabited (R ≃+* R)
• RingEquiv.Simps.symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : S → R
• RingEquiv.instEquivLike 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] : EquivLike (R ≃+* S) R S
• RingEquiv.toEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [Mul R] [Mul S] [Add R] [Add S] (self : R ≃+* S) : R ≃ S
• RingEquiv.ofUnique 📋 Mathlib.Algebra.Ring.Equiv
  {M : Type u_7} {N : Type u_8} [Unique M] [Unique N] [Add M] [Mul M] [Add N] [Mul N] : M ≃+* N
• RingEquiv.instUnique 📋 Mathlib.Algebra.Ring.Equiv
  {M : Type u_7} {N : Type u_8} [Unique M] [Unique N] [Add M] [Mul M] [Add N] [Mul N] : Unique (M ≃+* N)
• RingEquiv.toAddEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [Mul R] [Mul S] [Add R] [Add S] (self : R ≃+* S) : R ≃+ S
• RingEquiv.toMulEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [Mul R] [Mul S] [Add R] [Add S] (self : R ≃+* S) : R ≃* S
• RingEquiv.opOp 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_7) [Add R] [Mul R] : R ≃+* Rᵐᵒᵖᵐᵒᵖ
• RingEquiv.symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : S ≃+* R
• RingEquiv.symm_refl 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} [Mul R] [Add R] : (RingEquiv.refl R).symm = RingEquiv.refl R
• RingEquiv.cast 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} [(i : ι) → Mul (R i)] [(i : ι) → Add (R i)] {i j : ι} (h : i = j) : R i ≃+* R j
• RingEquiv.instRingEquivClass 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] : RingEquivClass (R ≃+* S) R S
• RingEquiv.symm_bijective 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] : Function.Bijective RingEquiv.symm
• RingEquivClass.toRingEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {F : Type u_1} {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [Mul β] [Add β] [EquivLike F α β] [RingEquivClass F α β] (f : F) : α ≃+* β
• RingEquiv.toNonUnitalRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) : R →ₙ+* S
• RingEquiv.op 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_7} {β : Type u_8} [Add α] [Mul α] [Add β] [Mul β] : α ≃+* β ≃ (αᵐᵒᵖ ≃+* βᵐᵒᵖ)
• RingEquiv.unop 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_7} {β : Type u_8} [Add α] [Mul α] [Add β] [Mul β] : αᵐᵒᵖ ≃+* βᵐᵒᵖ ≃ (α ≃+* β)
• RingEquiv.trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : R ≃+* S'
• RingEquiv.toNonUnitalRingHom_injective 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] : Function.Injective RingEquiv.toNonUnitalRingHom
• RingEquiv.symm_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : e.symm.symm = e
• RingEquiv.coe_refl 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_7) [Mul R] [Add R] : ⇑(RingEquiv.refl R) = id
• RingEquiv.refl_apply 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [Mul R] [Add R] (x : R) : (RingEquiv.refl R) x = x
• RingEquiv.toRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : R →+* S
• RingEquiv.toEquiv_eq_coe 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : f.toEquiv = ↑f
• RingEquiv.toOpposite 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [NonUnitalCommSemiring R] : R ≃+* Rᵐᵒᵖ
• RingEquiv.self_trans_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Add R] [Add S] [Mul R] [Mul S] (e : R ≃+* S) : e.trans e.symm = RingEquiv.refl R
• RingEquiv.symm_trans_self 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Add R] [Add S] [Mul R] [Mul S] (e : R ≃+* S) : e.symm.trans e = RingEquiv.refl S
• RingEquiv.toRingHom_injective 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] : Function.Injective RingEquiv.toRingHom
• RingEquiv.bijective 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : Function.Bijective ⇑e
• RingEquiv.injective 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : Function.Injective ⇑e
• RingEquiv.surjective 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : Function.Surjective ⇑e
• RingEquiv.ofBijective 📋 Mathlib.Algebra.Ring.Equiv
  {F : Type u_1} {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Bijective ⇑f) : R ≃+* S
• RingEquiv.toMonoidHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : R →* S
• RingEquiv.toAddMonoidHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : R →+ S
• RingEquiv.coe_addEquiv_refl 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [Mul R] [Add R] : ↑(RingEquiv.refl R) = AddEquiv.refl R
• RingEquiv.coe_mulEquiv_refl 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [Mul R] [Add R] : ↑(RingEquiv.refl R) = MulEquiv.refl R
• RingEquiv.coe_toEquiv_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : ↑e.symm = (↑e).symm
• 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.invFun_eq_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : EquivLike.inv f = ⇑f.symm
• RingEquiv.ofNonUnitalRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (hom : R →ₙ+* S) (inv : S →ₙ+* R) (hom_inv_id : inv.comp hom = NonUnitalRingHom.id R) (inv_hom_id : hom.comp inv = NonUnitalRingHom.id S) : R ≃+* S
• RingEquiv.toAddEquiv_eq_coe 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : f.toAddEquiv = ↑f
• RingEquiv.toMulEquiv_eq_coe 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : f.toMulEquiv = ↑f
• RingEquiv.symm_toNonUnitalRingHom_comp_toNonUnitalRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) : e.symm.toNonUnitalRingHom.comp e.toNonUnitalRingHom = NonUnitalRingHom.id R
• RingEquiv.toNonUnitalRingHomm_comp_symm_toNonUnitalRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) : e.toNonUnitalRingHom.comp e.symm.toNonUnitalRingHom = NonUnitalRingHom.id S
• RingEquiv.symm_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm
• RingEquiv.coe_toEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : ⇑↑f = ⇑f
• RingEquiv.ofRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) : R ≃+* S
• RingEquiv.piUnique 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (R : ι → Type u_8) [Unique ι] [(i : ι) → NonUnitalNonAssocSemiring (R i)] : ((i : ι) → R i) ≃+* R default
• RingEquiv.piMulOpposite 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (S : ι → Type u_8) [(i : ι) → NonUnitalNonAssocSemiring (S i)] : ((i : ι) → S i)ᵐᵒᵖ ≃+* ((i : ι) → (S i)ᵐᵒᵖ)
• AddEquiv.toRingEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} {F : Type u_9} [Add R] [Add S] [Mul R] [Mul S] [EquivLike F R S] [AddEquivClass F R S] (f : F) (H : ∀ (x y : R), f (x * y) = f x * f y) : R ≃+* S
• MulEquiv.toRingEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} {F : Type u_9} [Add R] [Add S] [Mul R] [Mul S] [EquivLike F R S] [MulEquivClass F R S] (f : F) (H : ∀ (x y : R), f (x + y) = f x + f y) : R ≃+* S
• RingEquiv.congr_arg 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] {f : R ≃+* S} {x x' : R} : x = x' → f x = f x'
• RingEquiv.apply_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (x : S) : e (e.symm x) = x
• RingEquiv.symm_apply_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (x : R) : e.symm (e x) = x
• 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.opOp_apply 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_7) [Add R] [Mul R] (a✝ : R) : (RingEquiv.opOp R) a✝ = MulOpposite.op (MulOpposite.op a✝)
• RingEquiv.coe_coe_toEquiv_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : ⇑(↑e).symm = ⇑e.symm
• RingEquiv.eq_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) {x : S} {y : R} : y = e.symm x ↔ e y = x
• RingEquiv.symm_apply_eq 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) {x : S} {y : R} : e.symm x = y ↔ x = e y
• RingEquiv.cast_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} [(i : ι) → Mul (R i)] [(i : ι) → Add (R i)] {i j : ι} (h : i = j) (a : R i) : (RingEquiv.cast h) a = cast ⋯ a
• RingEquiv.image_eq_preimage_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (s : Set R) : ⇑e '' s = ⇑e.symm ⁻¹' s
• RingEquiv.image_symm_eq_preimage 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (s : Set S) : ⇑e.symm '' s = ⇑e ⁻¹' s
• RingEquiv.symm_toRingHom_comp_toRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : e.symm.toRingHom.comp e.toRingHom = RingHom.id R
• RingEquiv.toRingHom_comp_symm_toRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : e.toRingHom.comp e.symm.toRingHom = RingHom.id S
• RingEquiv.congr_fun 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] {f g : R ≃+* S} (h : f = g) (x : R) : f x = g x
• RingEquiv.ext 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] {f g : R ≃+* S} (h : ∀ (x : R), f x = g x) : f = g
• RingEquiv.symm_toNonUnitalRingHom_apply_toNonUnitalRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (x : R) : e.symm.toNonUnitalRingHom (e.toNonUnitalRingHom x) = x
• RingEquiv.toNonUnitalRingHom_apply_symm_toNonUnitalRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (y : S) : e.toNonUnitalRingHom (e.symm.toNonUnitalRingHom y) = y
• RingEquiv.ext_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] {f g : R ≃+* S} : f = g ↔ ∀ (x : R), f x = g x
• RingEquiv.piCongrRight 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} {S : ι → Type u_9} [(i : ι) → NonUnitalNonAssocSemiring (R i)] [(i : ι) → NonUnitalNonAssocSemiring (S i)] (e : (i : ι) → R i ≃+* S i) : ((i : ι) → R i) ≃+* ((i : ι) → S i)
• RingEquiv.sumArrowEquivProdArrow 📋 Mathlib.Algebra.Ring.Equiv
  (α : Type u_2) (β : Type u_3) (R : Type u_4) [NonAssocSemiring R] : (α ⊕ β → R) ≃+* (α → R) × (β → R)
• RingEquiv.opOp_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_7) [Add R] [Mul R] (a✝ : Rᵐᵒᵖᵐᵒᵖ) : (RingEquiv.opOp R).symm a✝ = MulOpposite.unop (MulOpposite.unop a✝)
• RingEquiv.symm_toRingHom_apply_toRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (x : R) : e.symm.toRingHom (e.toRingHom x) = x
• RingEquiv.toRingHom_apply_symm_toRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (y : S) : e.toRingHom (e.symm.toRingHom y) = y
• RingEquiv.cast_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} [(i : ι) → Mul (R i)] [(i : ι) → Add (R i)] {i j : ι} (h : i = j) (a : R j) : (RingEquiv.cast h).symm a = cast ⋯ a
• RingEquiv.piCongrRight_refl 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} [(i : ι) → NonUnitalNonAssocSemiring (R i)] : (RingEquiv.piCongrRight fun i => RingEquiv.refl (R i)) = RingEquiv.refl ((i : ι) → R i)
• RingEquiv.prodCongr 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {R' : Type u_8} {S : Type u_9} {S' : Type u_10} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring S'] (f : R ≃+* R') (g : S ≃+* S') : R × S ≃+* R' × S'
• RingEquiv.symm_ofNonUnitalRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (g : S →ₙ+* R) (h₁ : g.comp f = NonUnitalRingHom.id R) (h₂ : f.comp g = NonUnitalRingHom.id S) : (RingEquiv.ofNonUnitalRingHom f g h₁ h₂).symm = RingEquiv.ofNonUnitalRingHom g f h₂ h₁
• RingEquiv.piCongrLeft'_symm 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_2} {β : Type u_3} {R : Type u_7} [NonUnitalNonAssocSemiring R] (e : α ≃ β) : (RingEquiv.piCongrLeft' (fun x => R) e).symm = RingEquiv.piCongrLeft' (fun i => R) e.symm
• RingEquiv.coe_toAddEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : ⇑↑f = ⇑f
• RingEquiv.coe_toMulEquiv 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (f : R ≃+* S) : ⇑↑f = ⇑f
• RingEquiv.toNonUnitalRingHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂).toNonUnitalRingHom = e₂.toNonUnitalRingHom.comp e₁.toNonUnitalRingHom
• RingEquiv.coe_ofBijective 📋 Mathlib.Algebra.Ring.Equiv
  {F : Type u_1} {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Bijective ⇑f) : ⇑(RingEquiv.ofBijective f hf) = ⇑f
• RingEquiv.map_add 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (x y : R) : e (x + y) = e x + e y
• RingEquiv.map_mul 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) (x y : R) : e (x * y) = e x * e y
• RingEquiv.ofBijective_apply 📋 Mathlib.Algebra.Ring.Equiv
  {F : Type u_1} {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Bijective ⇑f) (x : R) : (RingEquiv.ofBijective f hf) x = f x
• RingEquiv.map_zero 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) : f 0 = 0
• RingEquiv.coe_toNonUnitalRingHom' 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) : ⇑f.toNonUnitalRingHom = ⇑f
• RingEquiv.coe_coe_toAddEquiv_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : ⇑(↑e).symm = ⇑e.symm
• RingEquiv.coe_coe_toMulEquiv_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : ⇑(↑e).symm = ⇑e.symm
• RingEquiv.coe_toAddEquiv_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : ↑e.symm = (↑e).symm
• RingEquiv.coe_toMulEquiv_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Mul R] [Mul S] [Add R] [Add S] (e : R ≃+* S) : ↑e.symm = (↑e).symm
• RingEquiv.trans_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) : (e₁.trans e₂) a = e₂ (e₁ a)
• RingEquiv.ofRingHom_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) : (RingEquiv.ofRingHom f g h₁ h₂).symm = RingEquiv.ofRingHom g f h₂ h₁
• RingEquiv.coe_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁
• RingEquiv.map_eq_zero_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) {x : R} : f x = 0 ↔ x = 0
• RingEquiv.map_ne_zero_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) {x : R} : f x ≠ 0 ↔ x ≠ 0
• RingEquiv.piOptionEquivProd 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : Option ι → Type u_8} [(i : Option ι) → NonUnitalNonAssocSemiring (R i)] : ((i : Option ι) → R i) ≃+* R none × ((i : ι) → R (some i))
• 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.toRingHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂).toRingHom = e₂.toRingHom.comp e₁.toRingHom
• RingEquiv.symm_trans_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : S') : (e₁.trans e₂).symm a = e₁.symm (e₂.symm a)
• RingEquiv.toOpposite_apply 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [NonUnitalCommSemiring R] (r : R) : (RingEquiv.toOpposite R) r = MulOpposite.op r
• RingEquiv.ofNonUnitalRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (hom : R →ₙ+* S) (inv : S →ₙ+* R) (hom_inv_id : inv.comp hom = NonUnitalRingHom.id R) (inv_hom_id : hom.comp inv = NonUnitalRingHom.id S) (a : R) : (RingEquiv.ofNonUnitalRingHom hom inv hom_inv_id inv_hom_id) a = hom a
• RingEquiv.map_one 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : f 1 = 1
• 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.map_eq_one_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) {x : R} : f x = 1 ↔ x = 1
• RingEquiv.map_ne_one_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) {x : R} : f x ≠ 1 ↔ x ≠ 1
• RingEquiv.ofNonUnitalRingHom_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (hom : R →ₙ+* S) (inv : S →ₙ+* R) (hom_inv_id : inv.comp hom = NonUnitalRingHom.id R) (inv_hom_id : hom.comp inv = NonUnitalRingHom.id S) (a : S) : (RingEquiv.ofNonUnitalRingHom hom inv hom_inv_id inv_hom_id).symm a = inv a
• RingEquiv.ofRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) (a : R) : (RingEquiv.ofRingHom f g h₁ h₂) a = f a
• RingEquiv.toNonUnitalRingHom_eq_coe 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) : f.toNonUnitalRingHom = ↑f
• RingEquiv.coe_addEquiv_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ↑(e₁.trans e₂) = (↑e₁).trans ↑e₂
• RingEquiv.coe_mulEquiv_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ↑(e₁.trans e₂) = (↑e₁).trans ↑e₂
• RingEquiv.toOpposite_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  (R : Type u_4) [NonUnitalCommSemiring R] (r : Rᵐᵒᵖ) : (RingEquiv.toOpposite R).symm r = MulOpposite.unop r
• RingEquiv.ofRingHom_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) (a : S) : (RingEquiv.ofRingHom f g h₁ h₂).symm a = g a
• RingEquiv.piCongrRight_symm 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} {S : ι → Type u_9} [(i : ι) → NonUnitalNonAssocSemiring (R i)] [(i : ι) → NonUnitalNonAssocSemiring (S i)] (e : (i : ι) → R i ≃+* S i) : (RingEquiv.piCongrRight e).symm = RingEquiv.piCongrRight fun i => (e i).symm
• RingEquiv.map_pow 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] (f : R ≃+* S) (a : R) (n : ℕ) : f (a ^ n) = f a ^ n
• RingEquiv.piCongrLeft 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {ι' : Type u_8} (S : ι' → Type u_9) (e : ι ≃ ι') [(i : ι') → NonUnitalNonAssocSemiring (S i)] : ((i : ι) → S (e i)) ≃+* ((i : ι') → S i)
• RingEquiv.coe_ringHom_refl 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} [NonAssocSemiring R] : ↑(RingEquiv.refl R) = RingHom.id R
• RingEquiv.map_neg_one 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocRing R] [NonAssocRing S] (f : R ≃+* S) : f (-1) = -1
• RingEquiv.map_eq_neg_one_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocRing R] [NonAssocRing S] (f : R ≃+* S) {x : R} : f x = -1 ↔ x = -1
• RingEquiv.comp_ofBijective_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : Function.Bijective ⇑f) : f.comp ↑(RingEquiv.ofBijective f hf).symm = NonUnitalRingHom.id S
• RingEquiv.ofBijective_symm_comp 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : Function.Bijective ⇑f) : (↑(RingEquiv.ofBijective f hf).symm).comp f = NonUnitalRingHom.id R
• RingEquiv.toRingHom_eq_coe 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : f.toRingHom = ↑f
• RingEquiv.piCongrLeft' 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {ι' : Type u_8} (R : ι → Type u_9) (e : ι ≃ ι') [(i : ι) → NonUnitalNonAssocSemiring (R i)] : ((i : ι) → R i) ≃+* ((i : ι') → R (e.symm i))
• RingEquiv.op_apply_apply 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_7} {β : Type u_8} [Add α] [Mul α] [Add β] [Mul β] (f : α ≃+* β) (a✝ : αᵐᵒᵖ) : (RingEquiv.op f) a✝ = MulOpposite.op (f (MulOpposite.unop a✝))
• RingEquiv.coe_ringHom_ofRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) : ↑(RingEquiv.ofRingHom f g h₁ h₂) = f
• RingEquiv.map_neg 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R ≃+* S) (x : R) : f (-x) = -f x
• RingEquiv.coe_toNonUnitalRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) : ⇑↑f = ⇑f
• RingEquiv.piUnique_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (R : ι → Type u_8) [Unique ι] [(i : ι) → NonUnitalNonAssocSemiring (R i)] : ⇑(RingEquiv.piUnique R) = fun f => f default
• RingEquiv.op_apply_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_7} {β : Type u_8} [Add α] [Mul α] [Add β] [Mul β] (f : α ≃+* β) (a✝ : βᵐᵒᵖ) : (RingEquiv.op f).symm a✝ = MulOpposite.op (f.symm (MulOpposite.unop a✝))
• RingEquiv.op_symm_apply_apply 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_7} {β : Type u_8} [Add α] [Mul α] [Add β] [Mul β] (f : αᵐᵒᵖ ≃+* βᵐᵒᵖ) (a✝ : α) : (RingEquiv.op.symm f) a✝ = MulOpposite.unop (f (MulOpposite.op a✝))
• RingEquiv.piCongrRight_trans 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} {S : ι → Type u_9} {T : ι → Type u_10} [(i : ι) → NonUnitalNonAssocSemiring (R i)] [(i : ι) → NonUnitalNonAssocSemiring (S i)] [(i : ι) → NonUnitalNonAssocSemiring (T i)] (e : (i : ι) → R i ≃+* S i) (f : (i : ι) → S i ≃+* T i) : (RingEquiv.piCongrRight e).trans (RingEquiv.piCongrRight f) = RingEquiv.piCongrRight fun i => (e i).trans (f i)
• RingEquiv.piEquivPiSubtypeProd 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (p : ι → Prop) [DecidablePred p] (Y : ι → Type u_8) [(i : ι) → NonUnitalNonAssocSemiring (Y i)] : ((i : ι) → Y i) ≃+* ((i : { x // p x }) → Y ↑i) × ((i : { x // ¬p x }) → Y ↑i)
• RingEquiv.op_symm_apply_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_7} {β : Type u_8} [Add α] [Mul α] [Add β] [Mul β] (f : αᵐᵒᵖ ≃+* βᵐᵒᵖ) (a✝ : β) : (RingEquiv.op.symm f).symm a✝ = MulOpposite.unop (f.symm (MulOpposite.op a✝))
• RingEquiv.piUnique_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (R : ι → Type u_8) [Unique ι] [(i : ι) → NonUnitalNonAssocSemiring (R i)] : ⇑(RingEquiv.piUnique R).symm = uniqueElim
• RingEquiv.coe_toRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : ⇑↑f = ⇑f
• RingEquiv.piCongrRight_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : ι → Type u_8} {S : ι → Type u_9} [(i : ι) → NonUnitalNonAssocSemiring (R i)] [(i : ι) → NonUnitalNonAssocSemiring (S i)] (e : (i : ι) → R i ≃+* S i) (x : (i : ι) → R i) (j : ι) : (RingEquiv.piCongrRight e) x j = (e j) (x j)
• RingEquiv.coe_addMonoidHom_refl 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} [NonAssocSemiring R] : ↑(RingEquiv.refl R) = AddMonoidHom.id R
• RingEquiv.map_sub 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R ≃+* S) (x y : R) : f (x - y) = f x - f y
• RingEquiv.coe_nonUnitalRingHom_inj_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f g : R ≃+* S) : f = g ↔ ↑f = ↑g
• 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.sumArrowEquivProdArrow_apply 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_2} {β : Type u_3} {R : Type u_4} [NonAssocSemiring R] (x : α ⊕ β → R) : (RingEquiv.sumArrowEquivProdArrow α β R) x = (Equiv.sumArrowEquivProdArrow α β R) x
• 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
• RingEquiv.coe_prodCongr 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {R' : Type u_8} {S : Type u_9} {S' : Type u_10} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring S'] (f : R ≃+* R') (g : S ≃+* S') : ⇑(f.prodCongr g) = Prod.map ⇑f ⇑g
• RingEquiv.prodCongr_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {R' : Type u_8} {S : Type u_9} {S' : Type u_10} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring S'] (f : R ≃+* R') (g : S ≃+* S') (a✝ : R × S) : (f.prodCongr g) a✝ = Prod.map (⇑f) (⇑g) a✝
• RingEquiv.piOptionEquivProd_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : Option ι → Type u_8} [(i : Option ι) → NonUnitalNonAssocSemiring (R i)] (f : (a : Option ι) → R a) : RingEquiv.piOptionEquivProd f = (f none, fun a => f (some a))
• RingEquiv.coe_monoidHom_refl 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} [NonAssocSemiring R] : ↑(RingEquiv.refl R) = MonoidHom.id R
• RingEquiv.toMonoidHom_commutes 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : ↑↑f = (↑f).toMonoidHom
• RingEquiv.toEquiv_commutes 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : (↑f).toEquiv = (↑f).toEquiv
• RingEquiv.comp_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : (↑e).comp ↑e.symm = RingHom.id S
• RingEquiv.symm_comp 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : (↑e.symm).comp ↑e = RingHom.id R
• RingEquiv.toAddMonoidMom_commutes 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : (↑f).toAddMonoidHom = (↑f).toAddMonoidHom
• RingEquiv.toNonUnitalRingHom_commutes 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) : ↑↑f = ↑f
• RingEquiv.coe_ringHom_inj_iff 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {S : Type u_8} [NonAssocSemiring R] [NonAssocSemiring S] (f g : R ≃+* S) : f = g ↔ ↑f = ↑g
• RingEquiv.sumArrowEquivProdArrow_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {α : Type u_2} {β : Type u_3} {R : Type u_4} [NonAssocSemiring R] (x : (α → R) × (β → R)) : (RingEquiv.sumArrowEquivProdArrow α β R).symm x = (Equiv.sumArrowEquivProdArrow α β R).symm x
• RingEquiv.prodCongr_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_7} {R' : Type u_8} {S : Type u_9} {S' : Type u_10} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring S'] (f : R ≃+* R') (g : S ≃+* S') (a✝ : R' × S') : (f.prodCongr g).symm a✝ = Prod.map (⇑f.symm) (⇑g.symm) a✝
• RingEquiv.piOptionEquivProd_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {R : Option ι → Type u_8} [(i : Option ι) → NonUnitalNonAssocSemiring (R i)] (x : R none × ((a : ι) → R (some a))) (a : Option ι) : RingEquiv.piOptionEquivProd.symm x a = Option.rec x.1 (fun val => x.2 val) a
• RingEquiv.ofRingHom_coe_ringHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (g : S →+* R) (h₁ : (↑f).comp g = RingHom.id S) (h₂ : g.comp ↑f = RingHom.id R) : RingEquiv.ofRingHom (↑f) g h₁ h₂ = f
• RingEquiv.coe_ringHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁
• RingEquiv.piCongrLeft'_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {ι' : Type u_8} (R : ι → Type u_9) (e : ι ≃ ι') [(i : ι) → NonUnitalNonAssocSemiring (R i)] (f : (a : ι) → R a) (x : ι') : (RingEquiv.piCongrLeft' R e) f x = f (e.symm x)
• RingEquiv.piCongrLeft_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {ι' : Type u_8} (S : ι' → Type u_9) (e : ι ≃ ι') [(i : ι') → NonUnitalNonAssocSemiring (S i)] (a✝ : (i : ι) → S (e.symm.symm i)) (i : ι') : (RingEquiv.piCongrLeft S e) a✝ i = ⋯ ▸ a✝ (e.symm i)
• RingEquiv.piCongrLeft_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {ι' : Type u_8} (S : ι' → Type u_9) (e : ι ≃ ι') [(i : ι') → NonUnitalNonAssocSemiring (S i)] (a✝ : (i : ι') → S i) (i : ι) : (RingEquiv.piCongrLeft S e).symm a✝ i = a✝ (e i)
• RingEquiv.coe_addMonoidHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonUnitalNonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁
• RingEquiv.piEquivPiSubtypeProd_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (p : ι → Prop) [DecidablePred p] (Y : ι → Type u_8) [(i : ι) → NonUnitalNonAssocSemiring (Y i)] (f : (i : ι) → Y i) : (RingEquiv.piEquivPiSubtypeProd p Y) f = (fun x => f ↑x, fun x => f ↑x)
• RingEquiv.piCongrLeft'_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} {ι' : Type u_8} (R : ι → Type u_9) (e : ι ≃ ι') [(i : ι) → NonUnitalNonAssocSemiring (R i)] (f : (b : ι') → R (e.symm b)) (x : ι) : (RingEquiv.piCongrLeft' R e).symm f x = ⋯ ▸ f (e x)
• RingEquiv.coe_monoidHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁
• RingEquiv.piEquivPiSubtypeProd_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {ι : Type u_7} (p : ι → Prop) [DecidablePred p] (Y : ι → Type u_8) [(i : ι) → NonUnitalNonAssocSemiring (Y i)] (f : ((i : { x // p x }) → Y ↑i) × ((i : { x // ¬p x }) → Y ↑i)) (x : ι) : (RingEquiv.piEquivPiSubtypeProd p Y).symm f x = if h : p x then f.1 ⟨x, h⟩ else f.2 ⟨x, h⟩
• OrderRingIso.toRingEquiv 📋 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) : α ≃+* β
• OrderRingIso.instCoeOutRingEquiv 📋 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] : CoeOut (α ≃+*o β) (α ≃+* β)
• OrderRingIso.coe_ringEquiv_refl 📋 Mathlib.Algebra.Order.Hom.Ring
  (α : Type u_2) [Mul α] [Add α] [LE α] : (OrderRingIso.refl α).toRingEquiv = RingEquiv.refl α
• OrderRingIso.toRingEquiv_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.toRingEquiv = f.toRingEquiv
• 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.coe_toRingEquiv 📋 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.toRingEquiv = ⇑f
• OrderRingIso.trans_toRingEquiv 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [Mul γ] [Add γ] [LE γ] (f : α ≃+*o β) (g : β ≃+*o γ) : (f.trans g).toRingEquiv = f.trans g.toRingEquiv
• OrderRingIso.trans_toRingEquiv_aux 📋 Mathlib.Algebra.Order.Hom.Ring
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [Mul γ] [Add γ] [LE γ] (f : α ≃+*o β) (g : β ≃+*o γ) : ↑(f.trans g) = f.trans g.toRingEquiv
• 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
• RingHomInvPair.toRingEquiv 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] (σ : R₁ →+* R₂) (σ' : R₂ →+* R₁) [RingHomInvPair σ σ'] : R₁ ≃+* R₂
• RingHomInvPair.toRingEquiv_apply 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] (σ : R₁ →+* R₂) (σ' : R₂ →+* R₁) [RingHomInvPair σ σ'] (a : R₁) : (RingHomInvPair.toRingEquiv σ σ') a = σ a
• RingHomInvPair.toRingEquiv_symm_apply 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] (σ : R₁ →+* R₂) (σ' : R₂ →+* R₁) [RingHomInvPair σ σ'] (a : R₂) : (RingHomInvPair.toRingEquiv σ σ').symm a = σ' a
• RingHomSurjective.instToRingHomRingEquiv 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] (σ : R₁ ≃+* R₂) : RingHomSurjective ↑σ
• RingHomInvPair.of_ringEquiv 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] (e : R₁ ≃+* R₂) : RingHomInvPair ↑e ↑e.symm
• RingHomInvPair.of_ringEquiv_symm 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] (e : R₁ ≃+* R₂) : RingHomInvPair ↑e.symm ↑e
• RingEquiv.toSemilinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Defs
  {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) : R ≃ₛₗ[↑f] S
• RingEquiv.toSemilinearEquiv_apply 📋 Mathlib.Algebra.Module.Equiv.Defs
  {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) (a : R) : f.toSemilinearEquiv a = f a
• RingEquiv.toSemilinearEquiv_symm_apply 📋 Mathlib.Algebra.Module.Equiv.Defs
  {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) (a✝ : S) : f.toSemilinearEquiv.symm a✝ = f.invFun a✝
• RingEquiv.symm_toSemilinearEquiv_symm_apply 📋 Mathlib.Algebra.Module.Equiv.Defs
  {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) (x : R) : f.symm.toSemilinearEquiv.symm x = f x
• RingEquiv.moduleEndSelf 📋 Mathlib.Algebra.Module.LinearMap.End
  (R : Type u_1) [Semiring R] : Rᵐᵒᵖ ≃+* Module.End R R
• RingEquiv.moduleEndSelfOp 📋 Mathlib.Algebra.Module.LinearMap.End
  (R : Type u_1) [Semiring R] : R ≃+* Module.End Rᵐᵒᵖ R
• RingEquiv.moduleEndSelf_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  (R : Type u_1) [Semiring R] (s : Rᵐᵒᵖ) : (RingEquiv.moduleEndSelf R) s = DistribSMul.toLinearMap R R s
• RingEquiv.moduleEndSelfOp_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  (R : Type u_1) [Semiring R] (s : R) : (RingEquiv.moduleEndSelfOp R) s = DistribSMul.toLinearMap Rᵐᵒᵖ R s
• RingEquiv.moduleEndSelf_symm_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  (R : Type u_1) [Semiring R] (f : Module.End R R) : (RingEquiv.moduleEndSelf R).symm f = MulOpposite.op (f 1)
• RingEquiv.moduleEndSelfOp_symm_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  (R : Type u_1) [Semiring R] (f : Module.End Rᵐᵒᵖ R) : (RingEquiv.moduleEndSelfOp R).symm f = f 1
• addMonoidEndRingEquivInt 📋 Mathlib.Algebra.Module.Equiv.Basic
  (A : Type u_9) [AddCommGroup A] : AddMonoid.End A ≃+* Module.End ℤ A
• LinearEquiv.conjRingEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁ : Type u_9} {R₂ : Type u_10} {M₁ : Type u_13} {M₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (e : M₁ ≃ₛₗ[σ₁₂] M₂) : Module.End R₁ M₁ ≃+* Module.End R₂ M₂
• Module.compHom.toLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_9} {S : Type u_10} [Semiring R] [Semiring S] (g : R ≃+* S) : R ≃ₗ[R] S
• LinearEquiv.conjRingEquiv_apply_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁ : Type u_9} {R₂ : Type u_10} {M₁ : Type u_13} {M₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (e : M₁ ≃ₛₗ[σ₁₂] M₂) (f : M₁ →ₗ[R₁] M₁) (x : M₂) : (e.conjRingEquiv f) x = e (f (e.symm x))

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